# Riesz and Green energy on projective spaces

Austin Anderson  Department of Mathematics
Florida State University<EMAIL_ADDRESS>,  Maria Dostert‡ Department of
Mathematics
KTH Royal Institute of Technology, Stockholm, Sweden<EMAIL_ADDRESS>,  Peter
J. Grabner* Institute of Analysis and Number Theory
Graz University of Technology, Graz, Austria<EMAIL_ADDRESS>,  Ryan
W. Matzke*<EMAIL_ADDRESS>and  Tetiana A. Stepaniuk$\diamond$ Institute
of Mathematics
University of Lübeck, Germany;
Institute of Mathematics NAS of Ukraine<EMAIL_ADDRESS>
###### Abstract.

In this paper we study Riesz, Green and logarithmic energy on two-point
homogeneous spaces. More precisely we consider the real, the complex, the
quaternionic and the Cayley projective spaces. For each of these spaces we
provide upper estimates for the mentioned energies using determinantal point
processes. Moreover, we determine lower bounds for these energies of the same
order of magnitude.

###### 2020 Mathematics Subject Classification:

Primary: 31C12 60G55, Secondary: 42C10 33C47

*These authors are supported by the Austrian Science Fund FWF project F5503 part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”.

‡This author was partially supported by the Wallenberg AI, Autonomous Systems
and Software Program (WASP) funded by the Knut and Alice Wallenberg
Foundation.

$\diamond$This author was supported by the Alexander von Humboldt Foundation.

##  1\. Introduction

Motivated by classical potential theory (see, for instance [40]) discrete
energies of point sets on manifolds have been studied. More precisely, for a
given symmetric and lower semi-continuous kernel
$K:\Omega\times\Omega\to\mathbb{R}$ on a metric space $\Omega$ the _discrete
energy_ of a set $\omega_{N}=\\{x_{1},\ldots,x_{N}\\}\subset\Omega$ is given
by

$E_{K}(\omega_{N})=\sum_{\begin{subarray}{c}i,j=1\\\ i\neq
j\end{subarray}}^{N}K(x_{i},x_{j}).$

In rather general settings the empirical measures associated to minimizing
configurations of $E_{K}(\omega_{N})$ for $N\to\infty$ converge weakly to the
minimizing measure of the continuous energy

$I_{K}(\mu)=\iint\limits_{\Omega\times\Omega}K(x,y)\,d\mu(x)\,d\mu(y)$

amongst all Borel probability measures. For more details and a comprehensive
introduction to the subject we refer to [16].

For a sufficiently repulsive potential, one expects the minimizing
configurations of the discrete energy to be well-distributed, in some sense.
Perhaps the best-known example of such potentials are the classical Riesz
$s$-energies ($s>0$) for infinite compact $\Omega\subseteq\mathbb{R}^{d}$

$J_{s}(x,y):=\frac{1}{\|x-y\|^{s}}.$

It has been shown in [32] that, under rather general conditions, if
$s\geq\dim(\Omega)$, then the minimizers of $E_{J_{s}}$ are uniformly
distributed. This uniformity of minimizers does not hold in general for
$s<\dim(\Omega)$. However, due to the highly symmetric structure of the sphere
$\mathbb{S}^{d-1}$, one finds that for the Riesz potentials $J_{s}$ with
$0<s<d-1$ and the logarithmic potential

$J_{0}(x,y)=-\log(\|x-y\|),$

which is obtained by a limiting process for $s\to 0$, the continuous energies
$I_{J_{s}}$ and $I_{J_{0}}$ are uniquely minimized by the uniform measure on
the sphere $\sigma$, and the minimizers of the discrete energies are uniformly
distributed.

The minimal energy of $N$ points for the kernel $J_{s}$ ($s\geq 0$) on a space
$\Omega$,

$\mathcal{E}_{J_{s}}(\Omega,N)=\min_{\omega_{N}\subset\Omega}E_{J_{s}}(\omega_{N})$

has been investigated especially for the sphere $\mathbb{S}^{d-1}$, see for
instance [55, 54]. For $0<s<d-1$ it satisfies

$-C_{1}N^{1+\frac{s}{d-1}}\leq\mathcal{E}_{J_{s}}(\mathbb{S}^{d-1},N)-I_{J_{s}}(\sigma)N^{2}\leq-
C_{2}N^{1+\frac{s}{d-1}},$

where $C_{1}$ and $C_{2}$ are positive constants. The term
$I_{J_{s}}(\sigma)N^{2}$ of highest order reflects the fact that the empirical
measures of the discrete minimizers weakly tend to $\sigma$. It is conjectured
that a more precise asymptotic equation

$\mathcal{E}_{J_{s}}(\mathbb{S}^{d-1},N)=I_{J_{s}}(\sigma)N^{2}-CN^{1+\frac{s}{d-1}}+o(N^{1+\frac{s}{d-1}})$

holds, where the precise value of the constant $C$ is believed to reflect the
local structure of minimizing configuration. For more details we refer to [19]
and [32]. The conjectural values of the constant are related to zeta functions
of certain lattices, which relates the question to lattice energies on
Euclidean spaces.

Motivated by these results, as well as certain other recent works mentioned
below, we extend these results known for spheres to the projective spaces
$\mathbb{FP}^{d-1}$ over scalar domains $\mathbb{F}$ (the real or complex
numbers, the quaternions, or the octonions). These spaces together with the
spheres are the only compact connected two-point homogeneous spaces (see
[56]). On these projective spaces, we study the energies given by the chordal
Riesz $s$-kernels

$K_{s}(x,y)=\frac{1}{\rho(x,y)^{s}}=\frac{1}{\sin(\vartheta(x,y))^{s}}\quad\text{for
}s>0,$

and chordal logarithmic kernel

$K_{0}(x,y)=-\log(\rho(x,y))=-\log\sin(\vartheta(x,y))$

where $\rho$ and $\vartheta$ are the chordal and geodesic metrics,
respectively, discussed below. The case of $s<\dim(\Omega)$ is the subject of
classical potential theory (see, for instance [40]), the corresponding kernels
are called _singular_ , whereas the kernels for $s\geq\dim(\Omega)$ are called
_hypersingular_. Each of the projective spaces can be embedded in a
sufficiently high dimensional unit sphere, in which case the chordal distance
becomes the Euclidean distance, making these energies the natural
generalization of the classical Riesz and logarithmic energies on the sphere,
but without requiring the embedding itself.

The study of Riesz energies on projective spaces, particularly
$\mathbb{CP}^{d-1}$, has been a subject of recent interest. In[1, 4, 5], the
authors studied various potential theoretic properties of the logarithmic
energy on complex projective spaces. The expected Riesz and logarithmic
energies of zero sets of independent Gaussian polynomials on
$\mathbb{CP}^{d-1}$ (and more generally on Kähler manifolds) was determined in
[26], whereas in [9], the authors computed the expected energies for certain
determinantal point processes to find asymptotic upper bounds on the Riesz and
logarithmic energies. More qualitative properties of the minimizers of the
logarithmic and Riesz energies on the real and complex projective spaces were
studied in [22]. The authors found that the minimizers of these energies are
uniformly distributed, including the hypersingular case, and that these
minimizers approximate tight frames (acting as element on the real and complex
spheres). Moreover, as $s\rightarrow\infty$, the minimizers of the Riesz
energies approximate best packings on these spaces (i.e. frames with low
coherence). Both tight frames on real and complex spaces as well as best
packings on Grassmannians have applications to signal processing (see, e.g.,
[45, 38, 21]).

Another natural kernel to study on the projective spaces is the Green
function, $G(x,y)$, associated to the Laplace-Beltrami operator. The Green
function is a smooth potential, intrinsic to any Riemannian manifold, that
behaves similarly to a Riesz energy at short ranges. On the sphere and
projective spaces, it is in fact a function of distance only, making
computations much more feasible. The minimization of Green energies on compact
Riemannian manifolds was first studied in [7], where it was shown that the
continuous Green energy is uniquely minimized by the uniform measure and that
minimizers of the discrete Green energy are uniformly distributed. The
minimizers of the Green energy have more recently been shown to be well-
seperated in [25], and to have the optimal asymptotic quadratic Wasserstein
distance from the uniform measure in [51]. Upper bounds for the minimal
discrete Green energy on complex projective spaces were determined in [9]
using determinantal point processes different from the ones used in our paper.
This upper bound was of the optimal order, which was determined for general
compact Riemannian manifolds in [51].

###  1.1. Summary of Paper and Main Results

In Sections 1.2 and 1.3 we list some necessary notation and properties of
Jacobi polynomials, which we make extensive use of in this paper.

In order to make this paper (mostly) self-contained and gather the required
material, in Section 2, we cover the necessary background for harmonic
analysis on compact connected two-point homogeneous spaces, as well as some of
its consequences. In Section 2.5, we obtain explicit formulae for the Green
functions on the projective spaces. In Section 2.6, we show that the Riesz and
logarithmic kernels are strictly positive definite, and obtain the following
result:

######  Theorem 1.1.

The continuous logarithmic energy $I_{K_{0}}$, Green energy $I_{G}$, and Riesz
$s$-energies $I_{K_{s}}$, for $0<s<\dim(\mathbb{FP}^{d-1})$, are uniquely
minimized by the uniform measure $\sigma$.

Moreover, if $\\{\omega_{N}\\}_{N=2}^{\infty}$ is a sequence of minimizers for
the discrete energies $E_{K_{0}}$, $E_{G}$, or $E_{K_{s}}$, for
$0<s<\dim(\mathbb{FP}^{d-1})$, then $\\{\omega_{N}\\}_{N=2}^{\infty}$ is a
sequence of uniformly distributed point configurations.

We finish this section by discussing the properties of the heat kernel we use
to find lower bounds on the minimal discrete Green energy.

In Section 3, we define determinantal point processes given by rotation
invariant kernels on the projective spaces. The processes are defined by
projections to spaces of harmonic functions, thus they are called _harmonic
ensembles_ following [12].

In Section 4, we provide lower and upper asymptotic bounds on the minimal
discrete Riesz, logarithmic, and Green energies on the projective spaces,
cumulating in the following three results:

######  Theorem 1.2.

For each projective space $\mathbb{FP}^{d-1}$ and
$0<\\!s\\!<\\!\dim(\mathbb{FP}^{d-1}\\!)$ there exist positive constants
$C_{s},C_{s}^{\prime}$ such that for $N\geq 2$

$-C_{s}N^{1+\frac{s}{\dim(\mathbb{FP}^{d-1})}}\leq\mathcal{E}_{K_{s}}(\mathbb{FP}^{d-1},N)-I_{K_{s}}(\sigma)N^{2}\leq-
C_{s}^{\prime}N^{1+\frac{s}{\dim(\mathbb{FP}^{d-1})}}.$

######  Theorem 1.3.

For each projective space $\mathbb{FP}^{d-1}$, there exist positive constants
$C_{0},C_{0}^{\prime}$ such that for $N\geq 2$

$-C_{0}N\log(N)\leq\mathcal{E}_{K_{0}}(\mathbb{FP}^{d-1},N)-I_{K_{0}}(\sigma)N^{2}\leq-
C_{0}^{\prime}N\log(N).$

######  Theorem 1.4.

For each projective space $\mathbb{FP}^{d-1}$, there exist positive constants
$C_{G},C_{G}^{\prime}$ such that for $N\geq 2$

$-C_{G}N^{2-\frac{2}{\dim(\mathbb{FP}^{d-1})}}\leq\mathcal{E}_{G}(\mathbb{FP}^{d-1},N)\leq-
C_{G}^{\prime}N^{2-\frac{2}{\dim(\mathbb{FP}^{d-1})}},$

unless $\mathbb{FP}^{d-1}=\mathbb{RP}^{2}$, in which case

$-C_{G}N\log(N)\leq\mathcal{E}_{G}(\mathbb{FP}^{d-1},N)\leq-
C_{G}^{\prime}N\log(N).$

The order of the upper bounds for each of these results is proved in Section
4.1 through jittered sampling using equal area partitions with some extra
control on the diameters. Such partitions exist on general Ahlfors regular
metric measure spaces by [30]. However, without a deeper understanding of the
geometry of these spaces, the method of jittered sampling does not give
explicit values for the constants $C_{s}^{\prime}$, $C_{0}^{\prime}$, and
$C_{G}^{\prime}$ in the above theorems. In Section 4.2, we compute the
expected Riesz, logarithmic, and Green energies of the harmonic ensemble. This
provides a more concrete upper bound on the minimal energies, with an explicit
constant for the next-order term and the order of the error term, though only
for certain values of $N$ (see (4.5), (4.7), (4.8), and (4.9)). In addition,
we compute the expected Riesz $s$-energy for $s=\dim\mathbb{FP}^{d-1}$ of this
ensemble, resulting in an asymptotic upper bound on the minimum of this
hypersingular energy. In Section 4.3, we determine the order of the lower
bounds for the Riesz and logarithmic energies through linear programming using
the complete monotonicity of the corresponding kernels as a function of
chordal distance. Finally, in Section 4.4 we give lower bounds for the Green
energy, with explicit values of the next-order term. We achieve these lower
bounds again via linear programming, this time making use of lower bounds for
the Green function obtained from the positivity of the heat kernel.

We collect our explicit upper and lower bounds for the minimal discrete Green
energies on projective spaces, with the lower bounds holding for all $N$ and
the upper bound holding only for certain values of $N$, in Table 1 below.

$\Omega$ |  lower bound |  upper bound
---|---|---
$\mathbb{RP}^{2}$ |  $-\frac{1}{2}N\log(N)+\mathcal{O}(N)$ |  $-\frac{1}{2}N\log(N)+\mathcal{O}(N)$
$\mathbb{RP}^{3}$ |  $-\frac{3}{4}\left(\pi\right)^{\frac{1}{3}}N^{2-\frac{2}{3}}+\mathcal{O}(N\log(N))$ |  $-\frac{9}{16}\left(\frac{4}{3}\right)^{\frac{2}{3}}N^{2-\frac{2}{3}}+\mathcal{O}(N)$
$\mathbb{RP}^{d-1}$ |  $-\frac{d-1}{4(d-3)}\left(\frac{\sqrt{\pi}}{\Gamma(d/2)}\right)^{\frac{2}{d-1}}N^{2-\frac{2}{d-1}}$ $+\mathcal{O}(N^{2-\frac{3}{d-1}})$ |  $-\frac{(d-1)^{2}}{8(d-2)(d-3)}\left(\frac{\sqrt{\pi}}{\Gamma(\frac{d}{2})\Gamma(\frac{d+1}{2})}\right)^{\frac{2}{d-1}}$ $\times N^{2-\frac{2}{d-1}}+\mathcal{O}(N^{2-\frac{3}{d-1}})$
$\mathbb{CP}^{d-1}$ |  $-\frac{d-1}{4(d-2)}\big{(}\frac{1}{\Gamma(d)}\big{)}^{\frac{1}{d-1}}N^{2-\frac{2}{2d-2}}$ $+\mathcal{O}(N^{2-\frac{3}{2d-2}})$ |  $-\frac{(d-1)}{4(2d-3)}\left(\frac{1}{\Gamma(d)}\right)^{\frac{2}{d-1}}N^{2-\frac{2}{2d-2}}$ $+\mathcal{O}(N^{2-\frac{3}{2d-2}})$
$\mathbb{HP}^{d-1}$ |  $-\frac{d-1}{2(2d-3)}\big{(}\frac{1}{\Gamma(2d)}\big{)}^{\frac{1}{2d-2}}N^{2-\frac{2}{4d-4}}$ $+\mathcal{O}(N^{2-\frac{3}{4d-4}})$ |  $-\frac{(d-1)^{2}}{(2d-3)(4d-5)}\\!\left(\\!\frac{1}{\Gamma(2d)\Gamma(2d-1)}\\!\right)^{\frac{1}{2d-2}}$ $\times N^{2-\frac{2}{4d-4}}+\mathcal{O}(N^{2-\frac{3}{4d-4}})$
$\mathbb{OP}^{2}$ |  $-\frac{2}{7}\big{(}\frac{6}{11!}\big{)}^{\frac{1}{8}}N^{2-\frac{2}{16}}+\mathcal{O}(N^{2-\frac{3}{16}})$ |  $-\frac{16}{21}\left(\frac{6}{(11!)(8!)}\right)^{\frac{1}{8}}N^{2-\frac{2}{16}}$ $+\mathcal{O}(N^{2-\frac{3}{16}})$
Table 1. Lower and upper bounds for the minimal discrete Green energy on
projective spaces in terms of the number of points $N$. The upper bound only
holds for $N=\frac{(\alpha+\beta+2)_{n}(\alpha+2)_{n}}{(\beta+2)_{n}n!}$
(defined in 2.1)

For the complex projective spaces $\mathbb{CP}^{d-1}$ with $d>4$, this upper
bound is an improvement upon the previously best known upper bound

$\mathcal{E}_{G}(N)\leq-\frac{d-1}{4(d-2)}\Big{(}\frac{1}{(d-1)!}\Big{)}^{\frac{1}{d-1}}N^{2-\frac{2}{2d-2}}$

for $N=\binom{d+n-1}{n}$ [9, Theorem 1.3]. We note that in their paper,
Beltrán and Etayo took the volume of the $\mathbb{CP}^{d-1}$ to be
$\frac{\pi^{d-1}}{(d-1)!}$, so we have adjusted their result to match our
normalization (volume being $1$). This upper bound was achieved through a
determinantal point process on the complex space $\mathbb{C}^{d-1}$ with a
kernel constructed from functions on this space, and then mapping this process
to the complex projective space $\mathbb{CP}^{d-1}$ via the map
$z\mapsto(1,z)$, creating the projective ensemble. As pointed out, our upper
bound also comes from a determinantal point process, but the kernel in this
instance is built directly from functions on $\mathbb{CP}^{d-1}$. The
rotational invariance that results from this seems to lead to an improvement,
however, it also means that our bound holds for different values of $N$
($N=\frac{((d-1)!)^{2}}{n+1}\binom{d+n-1}{n}^{2}$).

Similarly, our estimates for the minimal Riesz and logarithmic energies on
complex projective spaces resulting from the harmonic ensemble ((4.5) and
(4.7)) generally match or improve upon previously known results. A first
instance of applying point processes to obtain estimates for the Riesz and
logarithmic energies on $\mathbb{CP}^{d-1}$ is [26]. There, Feng and Zelditch
studied the expectation of these energies for the zero set of $d-1$ degree $m$
Gaussian random polynomials. They obtain the correct main term and the correct
order of the second asymptotic term for $0\leq s\leq\min(4,D)$; nevertheless,
their second order term becomes positive for $s$ close to $4$.

More recently, Beltrán and Etayo also applied their projective ensemble to
find estimates for the minimal Riesz and logarithmic energies on
$\mathbb{CP}^{d-1}$. For $0<s<2d-2$ and $N=\binom{d+n-1}{n}$, they obtained
the upper bound [9, Theorem 3.3])

$\mathcal{E}_{K_{s}}(N)\leq
I_{K_{s}}(\sigma)N^{2}-\frac{(d-1)\Gamma\left(d-1-\frac{s}{2}\right)}{\left(\Gamma(d)\right)^{1-\frac{s}{2d-2}}}N^{1+\frac{s}{2d-2}}+o(N^{1+\frac{s}{2d-2}}).$

Numerical evidence strongly suggests that the coefficient of the next-order
term in this bound and the one we obtain in Theorem 4.5 equate at some unique
value $s=s_{d-1}$ in each dimension, with their bound winning for values below
$s_{d-1}$ and ours being better for values above $s_{d-1}$. These values
$s_{d-1}$ appear to be bounded from above by some constant $s^{*}$. Assuming
such an $s^{*}$ exists, we believe $s^{*}\approx 6.0365$, which we obtain by
solving

$e^{-\frac{s^{*}}{2}}\frac{\Gamma\left(1+s^{*}\right)}{\Gamma\left(1+\frac{s^{*}}{2}\right)^{2}}=1,$

which results from letting $d$ to tend to infinity in the ratio of the two
coefficients and assuming them to be equal. In particular, our bound appears
to represent an improvement for all $s$ exceeding a fixed value independent of
dimension.

We also provide the expected logarithmic energy of our harmonic ensemble on
$\mathbb{CP}^{d-1}$ in Theorem 4.7, achieving the same next order term,
$-\frac{1}{2d-2}N\log(N)$, as achieved by the expected logarithmic energies in
[9, Corollary 3.4] and [26, Corollary 1]. The lower bound we determine in
Theorem 4.12 shows that this is indeed the best possible coefficient for the
second order term.

###  1.2. Notation

Throughout the paper we will use the following notations:

  * •

The _Pochhammer symbol_

$(x)_{k}=x(x+1)\cdots(x+k-1)=\frac{\Gamma(x+k)}{\Gamma(x)},$

  * •

the _digamma function_

$\psi(x)=\frac{\Gamma^{\prime}(x)}{\Gamma(x)}=-\gamma-\frac{1}{x}+\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+x}\right),$

  * •

the _harmonic numbers_

$H_{k}=\sum_{\ell=1}^{k}\frac{1}{\ell}=\psi(k+1)+\gamma,$

  * •

where

$\gamma=-\Gamma^{\prime}(1)=\lim_{k\to\infty}H_{k}-\log(k)$

denotes the _Euler-Mascheroni constant_.

  * •

We will also make frequent use of the asymptotic relations

$\frac{\Gamma(n+x)}{\Gamma(n+y)}=n^{x-y}\left(1+\mathcal{O}\left(\frac{1}{n}\right)\right)\text{as
}n\to\infty$

and

$\binom{n+x}{n}=\frac{n^{x}}{\Gamma(x+1)}\left(1+\mathcal{O}\left(\frac{1}{n}\right)\right)\text{as
}n\to\infty.$

  * •

We will denote the set of finite Borel measures on a space $\Omega$ as
$\mathcal{B}(\Omega)$, the set of Borel probability measures as
$\mathbb{P}(\Omega)$, the set of finite signed Borel measures as
$\mathcal{M}(\Omega)$, and the set of finite signed Borel measures with total
mass zero, i.e. $\nu\in\mathcal{M}(\Omega)$ satisfying $\nu(\Omega)=0$, as
$\mathcal{Z}(\Omega)$.

###  1.3. Jacobi polynomials

The classical Jacobi polynomials will play a prominent role in this paper.
Thus we collect some basic facts about them. The _Jacobi polynomials_
$P_{n}^{(\alpha,\beta)}(t)$ are the orthogonal polynomials for the weight
function $(1-t)^{\alpha}(1+t)^{\beta}$ on the interval $[-1,1]$. Throughout
the paper we will use the substitution $t=\cos(2\vartheta)$. The measure is
then normalised and transformed to a measure on the interval
$[0,\frac{\pi}{2}]$ that we denote by

(1.1)
$d\nu^{(\alpha,\beta)}(\vartheta)=\frac{1}{\gamma_{\alpha,\beta}}\sin(\vartheta)^{2\alpha+1}\cos(\vartheta)^{2\beta+1}\,d\vartheta,$

where

$\gamma_{\alpha,\beta}=\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{2\Gamma(\alpha+\beta+2)}.$

The Jacobi polynomials can be given by Rodrigues’ formula (see [43, p. 213])

$P_{n}^{(\alpha,\beta)}(t)=\frac{(-1)^{n}}{2^{n}n!}\frac{1}{(1-t)^{\alpha}(1+t)^{\beta}}\frac{d^{n}}{dt^{n}}(1-t)^{n+\alpha}(1+t)^{n+\beta}.$

The value

$P_{n}^{(\alpha,\beta)}(1)=\binom{n+\alpha}{n}$

and the relation

$\int_{0}^{\frac{\pi}{2}}\left(P_{n}^{(\alpha,\beta)}(\cos(2\vartheta))\right)^{2}\,d\nu^{(\alpha,\beta)}(\vartheta)=\frac{\alpha+\beta+1}{2n+\alpha+\beta+1}\frac{(\alpha+1)_{n}(\beta+1)_{n}}{n!(\alpha+\beta+1)_{n}}$

will occur frequently throughout.

We will use the summation formula

(1.2)
$\sum_{k=0}^{n}\frac{2k+\alpha+\beta+1}{\alpha+\beta+1}\frac{(\alpha+\beta+1)_{k}}{(\beta+1)_{k}}P_{k}^{(\alpha,\beta)}(t)=\frac{(\alpha+\beta+2)_{n}}{(\beta+1)_{n}}P_{n}^{(\alpha+1,\beta)}(t)$

at several occasions; this is a special case of a connection formula for
Jacobi polynomials with different parameters given in [3, Theorem 7.1.3].

Furthermore the orthogonality of the Jacobi polynomials allows expanding
functions $F(\cos(2\vartheta))\in
L^{2}([0,\frac{\pi}{2}],d\nu^{(\alpha,\beta)})$ in terms of
$P_{n}^{(\alpha,\beta)}$:

(1.3) $F(t)=\sum_{n=0}^{\infty}\widehat{F}(n)P_{n}^{(\alpha,\beta)}(t),$

where

(1.4)
$\widehat{F}(n)=\frac{m_{n}}{\left(P_{n}^{(\alpha,\beta)}(1)\right)^{2}}\int\limits_{0}^{\frac{\pi}{2}}F(\cos(2\vartheta))P_{n}^{(\alpha,\beta)}(\cos(2\vartheta))\,d\nu^{(\alpha,\beta)}(\vartheta)$

and

$m_{n}=\frac{2n+\alpha+\beta+1}{\alpha+\beta+1}\frac{(\alpha+\beta+1)_{n}(\alpha+1)_{n}}{n!(\beta+1)_{n}}.$

The convergence of the series (1.3) is _a priori_ in the $L^{2}$-sense. In the
case that $F$ is continuous on $[-1,1]$ and all the coefficients
$\widehat{F}(n)$ are non-negative, Mercer’s theorem (see, for instance [27])
ensures absolute and uniform convergence.

##  2\. Harmonic analysis on two-point homogeneous spaces

###  2.1. Classification of two-point homogeneous spaces

We call a connected Riemannian manifold $(\Omega,p)$ _homogeneous_ if there is
a Lie group $G$ acting transitively on $\Omega$. This implies that $\Omega$ is
homeomorphic to the quotient space $G/G_{a}$, where $G_{a}:=\\{g\in G:ga=a\\}$
is the stabilizer of a point $a\in\Omega$. The choice of $a\in\Omega$ does not
matter in this instance, as all stabilizers are conjugate by transitivity. Let
$\vartheta_{p}$ be the metric induced by the metric tensor $p$. The metric
induces a volume form, which we normalize to obtain the normalized surface
measure $\sigma$. If $G$ acts by isometries this equals the measure induced by
the Haar-measure on $G$.

If $G$ is the isometry group of the homogeneous space $\Omega$, we call
$\Omega$ _two-point homogeneous_ if for all $x_{1},x_{2},y_{1},y_{2}\in\Omega$
with $\vartheta_{p}(x_{1},x_{2})=\vartheta_{p}(y_{1},y_{2})$, there is an
isometry $g\in G$ such that $gx_{i}=y_{i}$, $i=1,2$. All two-point homogeneous
Riemannian manifolds have been classified (see [36, Chapter I.4]). The
noncompact spaces are the Euclidean spaces $\mathbb{R}^{d}$, the real,
complex, and quaternionic hyperbolic spaces, and the hyperbolic analogue of
the Cayley plane [53]. The only compact connected two-point homogeneous
Riemannian manifolds are the real unit spheres $\mathbb{S}^{d-1}$, the real
projective spaces $\mathbb{RP}^{d-1}$, the complex projective spaces
$\mathbb{CP}^{d-1}$, the quaternionic projective spaces $\mathbb{HP}^{d-1}$,
and the Cayley projective plane $\mathbb{OP}^{2}$ (see [56]), with the
quotient representations given in [57, pp. 28-29]

$\displaystyle\mathbb{S}^{d-1}$
$\displaystyle\cong\mathrm{SO}(d)/\mathrm{SO}(d-1)$
$\displaystyle\mathbb{RP}^{d-1}$
$\displaystyle\cong\mathrm{O}(d)/\Big{(}\mathrm{O}(d-1)\times\mathrm{O}(1)\Big{)},$
$\displaystyle\mathbb{CP}^{d-1}$
$\displaystyle\cong\mathrm{U}(d)/\Big{(}\mathrm{U}(d-1)\times\mathrm{U}(1)\Big{)},$
$\displaystyle\mathbb{HP}^{d-1}$
$\displaystyle\cong\mathrm{Sp}(d)/\Big{(}\mathrm{Sp}(d-1)\times\mathrm{Sp}(1)\Big{)},$
$\displaystyle\mathbb{OP}^{2}$ $\displaystyle\cong F_{4}/\mathrm{Spin}(9).$

When talking about the projective spaces in general we denote the scalar
domain by $\mathbb{F}$.

Note that it suffices to consider $\mathbb{FP}^{d-1}$ for $d>2$ only, as
$\mathbb{FP}^{1}$ is isomorphic to the sphere
$\mathbb{S}^{\dim_{\mathbb{R}}(\mathbb{F})}$ (see [6, p. 170]), so those will
not be considered in what follows.

For each two-point homogeneous space with underlying scalar domain
$\mathbb{F}$, we associate parameters

(2.1)
$\alpha=(d-1)\frac{\dim_{\mathbb{R}}(\mathbb{F})}{2}-1,\;\;\;\;\beta=\begin{cases}\alpha,&\text{
for }\Omega=\mathbb{S}^{d-1};\\\
\frac{\dim_{\mathbb{R}}(\mathbb{F})}{2}-1,&\text{ for
}\Omega=\mathbb{FP}^{d-1}.\end{cases}$

The dependence of $\alpha$ and $\beta$ on the space and its dimension will be
clarified in Section 2.3. Furthermore, we denote $D=\dim(\Omega)=2\alpha+2$
the dimension of the space $\Omega$ as a real manifold.

From now on, $\Omega$ always refers to a two-point homogeneous space, equipped
with metric tensor $p$ and corresponding $G$-invariant probability measure
$\sigma$, i.e. the normalized uniform surface measure. We let $\vartheta$
denote the geodesic distance, normalized to take values in
$[0,\frac{\pi}{2\kappa}]$, where $\kappa=\frac{1}{2}$ or $\kappa=1$, if
$\Omega$ is a sphere or projective space, respectively. We can also define a
chordal metric $\rho$ on each of these spaces by

(2.2)
$\rho(x,y)=\sin(\kappa\vartheta(x,y))=\sqrt{\frac{1-\cos(2\kappa\vartheta(x,y))}{2}},\quad
x,y\in\Omega.$

Note that on the sphere, this is the Euclidean distance $\frac{1}{2}\|x-y\|$
in ambient space, and on the complex projective space this is also known as
the Fubini-Study metric.

Each projective space $\mathbb{FP}^{d-1}$ can be canonically embedded into the
unit sphere $\mathbb{S}^{\tilde{d}-1}$, where
$\tilde{d}=d(\alpha+2)=\frac{d(d-1)}{2}\dim_{\mathbb{R}}(\mathbb{F})+d$, so
that the chordal metric is equivalent to the Euclidean metric on this
embedding, which we will now show.

Let $\mathcal{H}(\mathbb{F}^{d})$ be the set of all Hermitian $d\times d$
matrices with entries in $\mathbb{F}$. We see that
$\mathcal{H}(\mathbb{F}^{d})$ is a linear space over $\mathbb{R}$ of dimension
$\tilde{d}$, equipped with the symmetric real-valued inner product

$\langle
A,B\rangle=\frac{1}{2}\mathrm{tr}(A\overline{B^{\mathsf{T}}}+B\overline{A^{\mathsf{T}}})=\mathrm{Re}(\mathrm{tr}(A\overline{B^{\mathsf{T}}}))=\mathrm{Re}\sum_{i,j=1}^{d}a_{i,j}\overline{b_{i,j}}$

and norm

(2.3)
$\|A\|_{\mathcal{H}(\mathbb{F}^{d})}=(\mathrm{tr}(A\overline{A^{\mathsf{T}}}))^{\frac{1}{2}}=\Big{(}\sum_{i,j=1}^{d}|a_{i,j}|^{2}\Big{)}^{\frac{1}{2}}.$

For $\mathbb{F}\neq\mathbb{O}$, the orthogonal projector
$\Pi_{x}\in\mathcal{H}(\mathbb{F}^{d})$ ($x\in\mathbb{F}^{d}$, $\|x\|=1$) onto
a one-dimensional subspace $x\mathbb{F}$, can be given by the matrix
$\Pi_{x}=(x_{i}\overline{x}_{j})_{1\leq i,j\leq d}$, with
$x=(x_{1},\ldots,x_{d})$. Thus, the projective space can be written as

(2.4)
$\mathbb{FP}^{d-1}\cong\\{\Pi\in\mathcal{H}(\mathbb{F}^{d}):\Pi^{2}=\Pi,\mathrm{Tr}(\Pi)=1\\}.$

The group of isometries $U(d,\mathbb{F})$ acts on these projectors by
$g(\Pi)=g\Pi g^{-1}$.

For the Cayley plane, a similar model (as well as a detailed discussion) is
given in [6, 28]. In this model, one defines the Cayley plane by

(2.5)
$\mathbb{OP}^{2}\cong\\{\Pi\in\mathcal{H}(\mathbb{O}^{3}):\Pi^{2}=\Pi,\mathrm{Tr}(\Pi)=1\\}.$

Each matrix can be written as $\Pi_{x}=(x_{i}\overline{x_{j}})_{1\leq i,j\leq
3}$, for a vector $x=(x_{1},x_{2},x_{3})\in\mathbb{O}^{3}$ with
$\|x\|^{2}=|x_{1}|^{2}+|x_{2}|^{2}+|x_{3}|^{2}=1$ and
$(x_{1}x_{2})x_{3}=x_{1}(x_{2}x_{3})$ [33, Lemma 14.90].

Equations (2.3), (2.4), and (2.5) show us that for any
$\Pi\in\mathbb{FP}^{d-1}$, as defined by the above models,

$\|\Pi\|_{\mathcal{H}(\mathbb{F}^{d})}^{2}=\mathrm{Tr}(\Pi^{2})=\mathrm{Tr}(\Pi)=1,$

so the projective spaces are submanifolds in the unit sphere

$\mathbb{FP}^{d-1}\subset\\{\Pi\in\mathcal{H}(\mathbb{F}^{d}):\|\Pi\|_{\mathcal{H}(\mathbb{F}^{d})}=1\\}\subset\mathcal{H}(\mathbb{F}^{d})\cong\mathbb{R}^{\tilde{d}}.$

This provides an embedding $A$ of $\mathbb{FP}^{d-1}$ into the sphere
$\mathbb{S}^{\tilde{d}-1}$. The chordal metric $\rho(\Pi_{1},\Pi_{2})$, for
$\Pi_{1},\Pi_{2}\in\mathbb{FP}^{d-1}$, is then defined as the Euclidean
distance in the embedding

$\begin{split}\rho(\Pi_{1},\Pi_{2})&=\sqrt{1-\langle\Pi_{1},\Pi_{2}\rangle}=\frac{1}{\sqrt{2}}\|\Pi_{1}-\Pi_{2}\|_{\mathcal{H}(\mathbb{F}^{d})}\\\
&=\frac{1}{\sqrt{2}}\|A(\Pi_{1})-A(\Pi_{2})\|.\end{split}$

###  2.2. The Laplace operator and its eigenfunctions

Let $\triangle_{\Omega}$ be the Laplace-Beltrami operator on $\Omega$ induced
by the Riemannian metric $p$, and let $0=\lambda_{0}<\lambda_{1}<\cdots$ be
the eigenvalues of $\triangle_{\Omega}$ and for each $k\in\mathbb{N}$, let
$V_{k}:=V(\triangle_{\Omega},\lambda_{k})$ be the corresponding eigenspace and
$m_{k}=\dim(V_{k})$ be the multiplicity of $\lambda_{k}$. Notice that we
follow the convention of geometry choosing the sign of the operator so that
the eigenvalues are non-negative.

We then have the following version of the Spectral Theorem [24, Chapter 3,
Theorem 1.3, and Remark 1.2]

######  Theorem 2.1.

The eigenvalues of $\triangle_{\Omega}$ can be arranged in increasing order
$0=\lambda_{0}<\lambda_{1}<\cdots$, where
$\lim_{k\rightarrow\infty}\lambda_{k}=\infty$, each eigenspace $V_{k}$ has
finite dimension, and

$L^{2}(\Omega,\sigma)=\overline{\bigoplus_{k=0}^{\infty}V_{k}}.$

For any Riemannian manifold $M$, we can use geodesic polar coordinates
$(r,\vartheta_{1},\ldots,\vartheta_{\dim_{\mathbb{R}}(M)-1})$ to parametrize a
sufficiently small neighborhood $U$ of any point $a$, giving $U$ a positive
orientation and having a radial component $r(x)$ which is the distance of
$x\in M$ from $a$, as described in [13, Chapter 2], [48, Chapter 2.4], and
[34, Chapters IX.5, X.4, and X.7.4].

In particular, on a two-point homogeneous space $\Omega$, such a polar
coordinate parameterization can be defined on
$\Omega_{a}:=\Omega\setminus\mathbb{S}_{\Omega}(a,\frac{\pi}{2\kappa})$, where
$\mathbb{S}_{\Omega}(a,r)=\\{x\in\Omega:\vartheta(a,x)=r\\}$ for
$r\in[0,\frac{\pi}{2\kappa}]$ [34, Chapters IX.5, X.4, and X.7.4]. This allows
us to separate $\triangle_{\Omega}$ into a radial and angular component on
this set.

######  Theorem 2.2 ([34, Chapter X.7.4, Lemma 7.12]).

If $f\in C^{\infty}(\Omega)$ and $a\in\Omega$, then on $\Omega_{a}$ the
Laplace operator can be expressed in terms of geodesic polar coordinates by

$\triangle_{\Omega}f=-\frac{1}{A(r)}\frac{\partial}{\partial
r}\left(A(r)\frac{\partial f}{\partial r}\right)+\triangle_{\vartheta}f,$

where $\triangle_{\vartheta}$ is the Laplace operator on
$\mathbb{S}_{\Omega}(a,r)$, and $A(r)$ denotes the surface measure of
$\mathbb{S}_{\Omega}(a,r)$.

For each $r\in[0,\frac{\pi}{2\kappa}]$, $\mathbb{S}_{\Omega}(a,r)$ is a
submanifold of $\Omega$ with Riemannian structure induced by that of $\Omega$.
In this case we have for $0<r<\frac{\pi}{2\kappa}$ (see [35, Proposition 5.6
and p. 171])

(2.6) $A(r)=c\kappa^{-2\alpha-1}\sin^{2\alpha+1}(\kappa
r)\cos^{2\beta+1}(\kappa r),$

where $c$ is a constant depending on the structure of $\Omega$, the values
$\alpha$ and $\beta$ are given by (2.1). For the spaces $\mathbb{FP}^{d-1}$ we
choose $\kappa=1$, whereas for $\mathbb{S}^{d-1}$ we set $\kappa=\frac{1}{2}$.

For functions $f$ only depending on $r$ the Laplace operator then becomes

$\triangle_{r}=-\frac{1}{\sin^{2\alpha+1}(\kappa r)\cos^{2\beta+1}(\kappa
r)}\frac{d}{dr}\left(\sin^{2\alpha+1}(\kappa r)\cos^{2\beta+1}(\kappa
r)\frac{d}{dr}\right),$

which, making the substitution $z=\cos(2\kappa r)$, becomes

$\triangle_{z}=-\frac{4\kappa^{2}}{(1-z)^{\alpha}(1+z)^{\beta}}\frac{d}{dz}\left((1-z)^{\alpha+1}(1+z)^{\beta+1}\frac{d}{dz}\right),$

[29, pp. 177-178]. This is the Jacobi operator, for which the only
eigenfunctions continuous on $[-1,1]$ are the Jacobi polynomials
$P^{(\alpha,\beta)}_{k}(z)$, with corresponding eigenvalues
$\lambda_{k}=4\kappa^{2}k(k+\alpha+\beta+1)$ (see [52, Theorem 4.2.2]).
Summing up this discussion we have shown

######  Theorem 2.3.

Let $\Omega$ be a two-point homogeneous space of diameter
$\frac{\pi}{2\kappa}$ and $a\in\Omega$. Then the eigenfunctions of
$\triangle_{\Omega}$ on $\Omega$ depending only on $\vartheta(a,x)$ are given
by

$cP_{k}^{(\alpha,\beta)}(\cos(2\kappa\vartheta(x,a)))\quad\text{with
}c\in\mathbb{R}$

and the corresponding eigenvalues are

$\lambda_{k}=4\kappa^{2}k(k+\alpha+\beta+1)$

with the values of $\alpha$ and $\beta$ given by (2.1).

The eigenvalues and the dimensions of their corresponding eigenspaces are
given in Table 2 (see [14, 20, 31, 49]).

$\Omega$ | $\alpha$ | $\beta$ | $\lambda_{k}$ | $m_{k}=\dim(V_{k})$
---|---|---|---|---
$\mathbb{S}^{d-1}$ | $\frac{d-3}{2}$ | $\frac{d-3}{2}$ | $k(k+d-2)$ | $\frac{2k+d-2}{d-2}\binom{k+d-3}{d-3}$
$\mathbb{RP}^{d-1}$ | $\frac{d-3}{2}$ | $-\frac{1}{2}$ | $2k(2k+d-2)$ | $\frac{4k+d-2}{d-2}\binom{2k+d-3}{d-3}$
$\mathbb{CP}^{d-1}$ | $d-2$ | $0$ | $4k(k+d-1)$ | $\frac{2k+d-1}{d-1}\binom{d+k-2}{d-2}^{2}$
$\mathbb{HP}^{d-1}$ | $2d-3$ | $1$ | $4k(k+2d-1)$ | $\frac{2k+2d-1}{(2d-1)(2d-2)}\binom{k+2d-2}{2d-2}\binom{k+2d-3}{2d-3}$
$\mathbb{OP}^{2}$ | $7$ | $3$ | $4k(k+11)$ | $\frac{2k+11}{1320}\binom{k+10}{7}\binom{k+7}{7}$
Table 2.  The eigenvalues and dimensions of the eigenspaces of the Laplace
operator for two-point homogeneous spaces

For all two-point homogeneous spaces, we have the general formula

$m_{k}=\frac{2k+\alpha+\beta+1}{\alpha+\beta+1}\frac{(\alpha+\beta+1)_{k}(\alpha+1)_{k}}{k!(\beta+1)_{k}}.$

###  2.3. Irreducibility of representations

Let $\Omega$ be a two-point homogeneous space, with isometry group $G$ and
$G_{a}$ the stabilizer of some $a\in\Omega$, meaning that $\Omega\simeq
G/G_{a}$. We then have an orthogonal representation of $G$ in the pre-Hilbert
space $C^{\infty}(\Omega)$ given by $g\mapsto f(gx)$. Let $V$ be a finite-
dimensional subspace of $C^{\infty}(\Omega)$, invariant under this
representation, and let $q_{V}$ be the representation induced in $V$.

######  Definition 2.4.

A function $f$ is called _zonal_ with respect to $a\in\Omega$ if $f(a)\neq 0$,
and for all $g\in G_{a}$ and $x\in\Omega$, $f(gx)=f(x)$. The zonal functions
of $V$ form a vector subspace of $V$, denoted by $Z_{a}(V)$.

######  Lemma 2.5.

If $V\neq\\{0\\}$, then $Z_{a}(V)\neq\\{0\\}$. If $\dim(Z_{a}(V))=1$ for one
(and thus all) $a\in\Omega$, $q_{V}$ is irreducible.

###### Proof.

Consider the linear map $\phi:V\rightarrow\mathbb{R}$ defined by

$\phi(f)=f(a).$

Since, by assumption, $V$ contains a non-zero function, $G$ is transitive on
$\Omega$, and $V$ is $G$-invariant there exists some $f\in V$ such that
$\phi(f)$ is not zero. Thus $\ker(\phi)$ is a subspace of $V$, which is
invariant under $G_{a}$. The orthogonal subspace $(\ker(\phi))^{\perp}$ is
also invariant under $G_{a}$. Now, let $f\in(\ker(\phi))^{\perp}$ and $\mu$ be
the Haar measure on $G_{a}$. Define $f^{*}(x)=\int_{G_{a}}f(hx)\,d\mu(h)$ for
all $x\in\Omega$. Then $f^{*}$ is in $V$, nonzero at $a$, and is
$G_{a}$-invariant, meaning it is a zonal function, proving our first
statement.

Suppose that $q_{V}$ is reducible. Then, since it is an orthogonal
representation, we have a decomposition

$V=V^{\prime}\oplus V^{\prime\prime}$

where $V^{\prime}$ and $V^{\prime\prime}$ are both $G$-invariant and
nontrivial. Then each of the two spaces contains a nontrivial subspace of
zonal functions, so $\dim(Z_{a}(V))\geq 2$. ∎

######  Proposition 2.6.

For each $k\in\mathbb{N}_{0}$, let $V_{k}$ be the space of eigenfunctions of
$\triangle_{\Omega}$ associated to the $k$-th eigenvalue $\lambda_{k}$, and
let $q_{k}$ be the orthogonal representation of $G$ in $V_{k}$. The zonal
functions of $V_{k}$ are exactly
$cP^{(\alpha,\beta)}_{k}(\cos(2\kappa\vartheta(x,a)))$ for $c\in\mathbb{R}$,
and so $q_{k}$ is irreducible.

###### Proof.

By Lemma 2.5, $Z_{a}(V_{k})\neq\\{0\\}$, and it suffices to show that
$\dim(Z_{a}(V_{k}))\leq 1$. Let $f$ be a zonal function. Then two-point
homogeneity and Theorem 2.3 tell us that
$f(x)=cP_{k}^{(\alpha,\beta)}(\cos(2\kappa\vartheta(x,a)))$ for some
$c\in\mathbb{R}$. Thus the space $Z_{a}(V_{k})$ is one-dimensional, and the
proposition is proved. ∎

As a consequence of Theorem 2.1 and Proposition 2.6 we have

######  Proposition 2.7.

Let $H$ be a finite-dimensional $G$-invariant subspace of
$L^{2}(\Omega,\sigma)$. Then there exist $0\leq k_{1}<\cdots<k_{m}$ such that

$H=V_{k_{1}}\oplus\cdots\oplus V_{k_{m}},$

where $V_{k}$ is the eigenspace of $\triangle_{\Omega}$ corresponding to the
eigenvalue $\lambda_{k}$.

###  2.4. The addition formula for eigenfunctions of the Laplace operator

For each $k\in\mathbb{N}_{0}$, $Y_{k,1},\ldots,Y_{k,m_{k}}$ be real-valued
functions and form an orthonormal basis of $V_{k}$, under the inner product
$\langle Y,Z\rangle=\int_{\Omega}Y(x)Z(x)\,d\sigma(x)$.

######  Theorem 2.8 (Addition Formula).

For each $k\in\mathbb{N}_{0}$ and $x,y\in\Omega$,

(2.7)
$\sum_{m=1}^{m_{k}}Y_{k,m}(x)Y_{k,m}(y)=\frac{m_{k}}{P^{(\alpha,\beta)}_{k}(1)}P^{(\alpha,\beta)}_{k}(\cos(2\kappa\vartheta(x,y))).$

###### Proof.

Let

$Y_{k}(x):=\begin{pmatrix}Y_{k,1}(x)\\\ \vdots\\\
Y_{k,m_{k}}(x)\end{pmatrix},$

and note that
$\sum_{m=1}^{m_{k}}Y_{k,m}(x)Y_{k,m}(y)=Y_{k}(x)^{\mathsf{T}}Y_{k}(y)$. For
any $\gamma\in G$, $Y_{k,1}(\gamma x),\ldots,Y_{k,m_{k}}(\gamma x)$ is also an
orthonormal basis of $V_{k}$, thus there is a
$q_{k}(\gamma)\in\mathrm{O}(V_{k})$, so that

$\displaystyle Y_{k}(\gamma x)^{\mathsf{T}}Y_{k}(\gamma y)$
$\displaystyle=(q_{k}(\gamma)Y_{k}(x))^{\mathsf{T}}q_{k}(\gamma)Y_{k}(y)$
$\displaystyle=Y_{k}(x)^{\mathsf{T}}q_{k}(\gamma)^{\mathsf{T}}q_{k}(\gamma)Y_{k}(y)=Y_{k}(x)^{\mathsf{T}}Y_{k}(y).$

In particular, this means that for all $\gamma\in G_{y}$,

$Y_{k}(\gamma x)^{\mathsf{T}}Y_{k}(y)=Y_{k}(\gamma x)^{\mathsf{T}}Y_{k}(\gamma
y)=Y_{k}(x)^{\mathsf{T}}Y_{k}(y)$

making $Y_{k}(x)^{\mathsf{T}}Y_{k}(y)$ a zonal function of $x$, thus it must
be a multiple of $P^{(\alpha,\beta)}_{k}(\cos(2\vartheta(x,y)))$:

$Y_{k}(x)^{\mathsf{T}}Y_{k}(y)=c_{k}P_{k}^{(\alpha,\beta)}(\cos(2\kappa\vartheta(x,y))).$

Setting $x=y$ and integrating yields

$\sum_{m=1}^{m_{k}}\int_{\Omega}Y_{k,m}(x)^{2}\,d\sigma(x)=m_{k}=c_{k}P_{k}^{(\alpha,\beta)}(1),$

which gives (2.7). ∎

We note that one can achieve such an addition formula without making use of
properties of the Laplace operator and its eigenfunctions, and rather making
use of representation theory (see [23, Chapter 9.2.1], [42, Section 3.2]).

######  Remark 2.9.

From now on we only consider the projective spaces $\mathbb{RP}^{d-1}$,
$\mathbb{CP}^{d-1}$, $\mathbb{HP}^{d-1}$, and $\mathbb{OP}^{2}$; the case of
the sphere $\mathbb{S}^{d-1}$ has been treated in a very similar manner in
[12]. Thus we set $\kappa=1$ for the remaining part of the paper.

As a corollary of the area formula (2.6), we can see that the weighted measure
$\nu^{(\alpha,\beta)}$ given by (1.1) is related to integration on $\Omega$ in
the following way: for any $a\in\Omega$, and $F(\cos(2\vartheta))\in
L^{1}([0,\frac{\pi}{2}],\nu^{\alpha,\beta})$,

(2.8)
$\int\limits_{\Omega}F(\cos(2\vartheta(x,a)))\,d\sigma(x)=\int\limits_{0}^{\frac{\pi}{2}}F(\cos(2\vartheta))\,d\nu^{\alpha,\beta}(\vartheta).$

###  2.5. The Green function

The _Green function_ $G$ of the Laplace-Beltrami operator is given by the
integral operator solving the equation

$\triangle_{\Omega}g=f$

by

$g(x)=\int_{\Omega}G(x,y)f(y)\,d\sigma(y)$

with the additional condition $\int_{\Omega}g(x)\,d\sigma(x)=0$. In the case
of two-point homogeneous spaces the bivariate function $G(x,y)$ actually only
depends on the distance between $x$ and $y$ (see [7]). In fact, as we will
show below, the Green functions on the projective spaces have the following
closed form expressions.

######  Proposition 2.10.

The Green function for a projective space $\Omega$ is given by

$\begin{split}G(x,y)=&\frac{1}{4(\alpha+\beta+1)}\Bigg{(}\sum_{\ell=1}^{\alpha}\frac{\binom{\alpha}{\ell}(\ell-1)!}{(\beta+\alpha+1-\ell)_{\ell}}\frac{1}{\rho(x,y)^{2\ell}}-2\log(\rho(x,y))\\\
&-\psi(\alpha+\beta+1)-\psi(\alpha+\beta+2)-\gamma+\psi(\beta+1)\Bigg{)}\end{split}$

for $\alpha\in\mathbb{N}_{0}$ and

$\begin{split}G(x,y)=&\frac{\cos(\vartheta(x,y))}{2\alpha+1}\left(\frac{\pi}{2}-\vartheta(x,y)\right)\sum_{\ell=0}^{\alpha-\frac{1}{2}}\frac{(\frac{1}{2})_{\ell}}{\ell!}\frac{1}{\rho(x,y)^{2\ell+1}}\\\
&+\frac{1}{4\alpha+2}\sum_{\ell=1}^{\alpha-\frac{1}{2}}\frac{1}{\ell}\left(\frac{(\alpha+1-\ell)_{\ell}}{(\alpha+\frac{1}{2}-\ell)_{\ell}}-\frac{(\frac{1}{2})_{\ell}}{\ell!}\right)\frac{1}{\rho(x,y)^{2\ell}}\\\
&-\frac{1}{2\alpha+1}H_{\alpha-\frac{1}{2}}-\frac{1}{(2\alpha+1)^{2}}\end{split}$

for $\alpha\in\frac{1}{2}+\mathbb{N}_{0}$.

###### Proof.

Using the addition formula (2.7) we obtain the _formal_ expression

$\begin{split}&G(x,y)\\\
=&\sum_{n=1}^{\infty}\frac{2n+\alpha+\beta+1}{4(\alpha+\beta+1)n(n+\alpha+\beta+1)}\frac{(\alpha+\beta+1)_{n}}{(\beta+1)_{n}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y))).\end{split}$

Indeed, this series converges absolutely only for $\alpha<\frac{1}{2}$,
converges conditionally for $\frac{1}{2}\leq\alpha<\frac{3}{2}$, and does not
converge at all for $\alpha\geq\frac{3}{2}$. We will show that the series
converges in the sense of Abel’s summation method (see [58]) and we will
compute the limit.

We start with [3, p. 301, eq. (6.4.7)]

$\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_{n}}{(\beta+1)_{n}}P_{n}^{(\alpha,\beta)}(t)z^{n}\\\
=\frac{1}{(1+z)^{\alpha+\beta+1}}{\sideset{{}_{2}}{{}_{1}}{\operatorname{F}}\\!\left(\genfrac{}{}{0.0pt}{0}{\frac{\alpha+\beta+1}{2},\frac{\alpha+\beta+2}{2}}{\beta+1}\middle|\,\frac{2z(1+t)}{(1+z)^{2}}\right)}$

and define

$\begin{split}&\mathcal{G}_{\alpha,\beta}(\cos(2\vartheta))=\frac{1}{\eta}\lim_{z\to
1}\sum_{n=1}^{\infty}\frac{2n+\eta}{4n(n+\eta)}\frac{(\eta)_{n}}{(\beta+1)_{n}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta))z^{n}\\\
=&\int\limits_{0}^{1}\\!\left(\frac{1}{(1+z)^{\eta}}{\sideset{{}_{2}}{{}_{1}}{\operatorname{F}}\\!\left(\genfrac{}{}{0.0pt}{0}{\frac{\eta}{2},\frac{\eta+1}{2}}{\beta+1}\middle|\,\frac{4\cos(\vartheta)^{2}z}{(1+z)^{2}}\right)}-1\\!\right)\frac{1+z^{\eta}}{4\eta
z}\,dz,\end{split}$

where the convergence of the integral shows the existence of the limit for
$|\cos(\vartheta)|<1$. Here we have set $\eta=\alpha+\beta+1$ for short, which
we will use for this computation in the sequel.

The coefficient of $\cos(\vartheta)^{2n}$ for $n\geq 1$ is given by

$\frac{(\eta)_{2n}}{(\beta+1)_{n}n!}\int_{0}^{1}\frac{z^{n}}{(1+z)^{2n+\eta}}\frac{1+z^{\eta}}{4\eta
z}\,dz,$

where we have used

$4^{n}\left(\frac{\eta}{2}\right)_{n}\left(\frac{\eta+1}{2}\right)_{n}=(\eta)_{2n}.$

The integral is split into two parts; we first use the substitution
$z=\frac{1}{t}$ to obtain

$\int_{0}^{1}\frac{z^{n+\eta-1}}{(1+z)^{2n+\eta}}\,dz=\int_{1}^{\infty}\frac{t^{n-1}}{(1+t)^{2n+\eta}}\,dt.$

This shows that (see [43, p. 6])

$\int_{0}^{1}\frac{z^{n}}{(1+z)^{2n+\eta}}\frac{1+z^{\eta}}{4\eta
z}\,dz=\frac{1}{4\eta}\int_{0}^{\infty}\frac{z^{n-1}}{(1+z)^{2n+\eta}}\,dz=\frac{(n-1)!}{4\eta(n+\eta)_{n}}.$

For the constant term in the Taylor expansion we obtain

$C_{\eta}=-\int_{0}^{1}\left(1-\frac{1}{(1+z)^{\eta}}\right)\frac{1+z^{\eta}}{4\eta
z}\,dz=-\frac{1}{4\eta}\left(\psi(\eta+1)+\gamma\right).$

This can be obtained by splitting the integral in a similar way as above with
some slight modification to preserve convergence.

Putting everything together yields

$\begin{split}\mathcal{G}_{\alpha,\beta}(\cos(2\vartheta))=-\frac{1}{4(\alpha+\beta+1)}\Big{(}&\gamma+\psi(\alpha+\beta+2)\\\
&+\sum_{n=1}^{\infty}\frac{(\alpha+\beta+1)_{n}}{n(\beta+1)_{n}}\cos(\vartheta)^{2n}\Big{)}.\end{split}$

The derivation up to now was valid for all $\alpha,\beta>-1$. For the specific
values of $\alpha$ and $\beta$ occurring in the context of projective spaces
it turns out that the functions $\mathcal{G}_{\alpha,\beta}$ are indeed
elementary. For an alternative derivation of the Green function on the spaces
$\mathbb{FP}^{d-1}$ we refer to [7, Appendix A.1]

For $\alpha=k\in\mathbb{N}_{0}$ this is immediate from the observation that

$\frac{(\alpha+\beta+1)_{n}}{n(\beta+1)_{n}}=\frac{(n+\beta+1)_{k}}{n(\beta+1)_{k}},$

which is a polynomial in $n$ of degree $k$ divided by $n$. More precisely, we
have

$(n+\beta+1)_{k}=\sum_{\ell=0}^{k}\binom{k}{\ell}(\beta+1)_{k-\ell}(n)_{\ell},$

from which we derive

$\displaystyle\sum_{n=1}^{\infty}\frac{(\alpha+\beta+1)_{n}\cos(\vartheta)^{2n}}{n(\beta+1)_{n}}$
$\displaystyle=$
$\displaystyle\frac{1}{(\beta+1)_{k}}\sum_{\ell=0}^{k}\binom{k}{\ell}(\beta+1)_{k-\ell}\sum_{n=1}^{\infty}\frac{(n)_{\ell}}{n}\cos(\vartheta)^{2n}$
$\displaystyle=$
$\displaystyle\sum_{\ell=1}^{k}\frac{\binom{k}{\ell}(\ell-1)!}{(\beta+k+1-\ell)_{\ell}}\frac{1}{\sin(\vartheta)^{2\ell}}+\log\frac{1}{\sin(\vartheta)^{2}}$
$\displaystyle-$ $\displaystyle\left(\psi(\beta+k+1)-\psi(\beta+1)\right).$

For $\alpha\in\mathbb{N}_{0}$ this gives

$\displaystyle\begin{split}&\mathcal{G}_{\alpha,\beta}(\cos(2\vartheta))=\frac{1}{4(\alpha+\beta+1)}\Bigg{(}\sum_{\ell=1}^{\alpha}\binom{\alpha}{\ell}\frac{(\ell-1)!}{(\beta+\alpha+1-\ell)_{\ell}}\frac{1}{\sin(\vartheta)^{2\ell}}-\gamma\\\
&+\log\frac{1}{\sin(\vartheta)^{2}}-\psi(\alpha+\beta+1)-\psi(\alpha+\beta+2)-\psi(\beta+1)\Bigg{)}.\end{split}$

The case that $\alpha\in\frac{1}{2}+\mathbb{N}_{0}$ only occurs in the case of
real projective spaces, when $\beta=-\frac{1}{2}$. In this case we consider
the functions

$\displaystyle F_{k}(z)$
$\displaystyle=\sum_{n=1}^{\infty}\frac{(k+1)_{n}}{n(\frac{1}{2})_{n}}z^{n}$
$\displaystyle G_{k}(z)$
$\displaystyle=2(k+1)\sum_{n=0}^{\infty}\frac{(k+2)_{n}}{(\frac{3}{2})_{n}}z^{n}=2(k+1){\sideset{{}_{2}}{{}_{1}}{\operatorname{F}}\\!\left(\genfrac{}{}{0.0pt}{0}{1,k+2}{\frac{3}{2}}\middle|\,z\right)};$

these satisfy $F_{k}^{\prime}=G_{k}$. From [47, p. 463, eqns. 132, 133] we
infer that

$G_{k}(z)=\frac{(\frac{3}{2})_{k}}{k!}\frac{\arcsin(\sqrt{z})}{\sqrt{z}}\frac{1}{(1-z)^{k+\frac{3}{2}}}+\sum_{\ell=0}^{k}\frac{(k+\frac{3}{2}-\ell)_{\ell}}{(k+1-\ell)_{\ell}}\frac{1}{(1-z)^{\ell+1}}.$

From this we compute

$F_{k}(z)=\frac{2\sqrt{z}\arcsin(\sqrt{z})}{\sqrt{1-z}}\sum_{\ell=0}^{k}\frac{(\frac{1}{2})_{\ell}}{\ell!}\frac{1}{(1-z)^{\ell}}\\\
+\sum_{\ell=1}^{k}\frac{1}{\ell}\left(\frac{(k+\frac{3}{2}-\ell)_{\ell}}{(k+1-\ell)_{\ell}}-\frac{(\frac{1}{2})_{\ell}}{\ell!}\right)\frac{1}{(1-z)^{\ell}}-H_{k},$

which can be checked by differentiation. We get

$\begin{split}\mathcal{G}_{k+\frac{1}{2},-\frac{1}{2}}(\cos(2\vartheta))=&\frac{1}{2(k+1)}\cos(\vartheta)\left(\frac{\pi}{2}-\vartheta\right)\sum_{\ell=0}^{k}\frac{(\frac{1}{2})_{\ell}}{\ell!}\frac{1}{\sin(\vartheta)^{2\ell+1}}\\\
&+\frac{1}{4(k+1)}\sum_{\ell=1}^{k}\frac{1}{\ell}\left(\frac{(k+\frac{3}{2}-\ell)_{\ell}}{(k+1-\ell)_{\ell}}-\frac{(\frac{1}{2})_{\ell}}{\ell!}\right)\frac{1}{\sin(\vartheta)^{2\ell}}\\\
&-\frac{1}{2(k+1)}H_{k}-\frac{1}{4(k+1)^{2}}.\end{split}$

∎

Putting everything together we derive the asymptotic main term of the Green
function

(2.9)
$\mathcal{G}_{\alpha,\beta}(\cos(2\vartheta))=\frac{\Gamma(\alpha)\Gamma(\beta+1)}{4\Gamma(\alpha+\beta+2)}\frac{1}{\sin(\vartheta)^{2\alpha}}+\mathcal{O}\left(\frac{1}{\sin(\vartheta)^{2\alpha-1}}\right),$

for $\alpha>0$ and

(2.10)
$\mathcal{G}_{0,-\frac{1}{2}}(\cos(2\vartheta))=-\log(\sin(\vartheta))+\mathcal{O}\left(1\right),$

for $\Omega=\mathbb{RP}^{2}$.

###  2.6. Minimizers of the Riesz, Green, and logarithmic Energies

In this section, we prove that the uniform measure on $\Omega$ induced by the
Haar measure of the group acting on $\Omega$ minimizes the Riesz, Green and
logarithmic energies.

######  Theorem 2.11.

The logarithmic energy $I_{K_{0}}$, the Green energy $I_{G}$, and the Riesz
$s$-energies $I_{K_{s}}$, for $0<s<2\alpha+2$, are uniquely minimized by the
uniform measure $\sigma$, with

(2.11)
$I_{K_{s}}(\sigma)=\frac{\Gamma(\alpha+\beta+2)\Gamma(\alpha+1-\frac{s}{2})}{\Gamma(\alpha+1)\Gamma(\alpha+\beta+2-\frac{s}{2})},$

(2.12) $I_{K_{0}}(\sigma)=\frac{1}{2}(\psi(\alpha+\beta+2)-\psi(\alpha+1)),$

and

(2.13) $I_{G}(\sigma)=0.$

Moreover, if $\\{\omega_{N}\\}_{N=2}^{\infty}$ is a sequence of minimizers for
the discrete energies $E_{K_{0}}$, $E_{K_{s}}$, or $E_{G}$, then the
normalized counting measures

$\nu_{\omega_{N}}=\frac{1}{N}\sum_{j=1}^{N}\delta_{z_{j}},$

converge weakly to $\sigma$.

We note that this has already been proven for Green energies in [7, Theorem
1.1], and the uniformity of minimizers for logarithmic and Riesz energies on
$\mathbb{RP}^{d-1}$ and $\mathbb{CP}^{d-1}$ was proven in [22, Theorem 4.1].

In general, the minimal discrete and continuous energies of some kernel $K$
are related in the following way

######  Theorem 2.12 ([16, Theorem 4.2.2]).

Let $K$ be a lower semi-continuous kernel on $\Omega$, and $\mu$ be a
minimizer of $I_{K}$. Then for all $N\in\mathbb{N}\setminus\\{1\\}$,

$\mathcal{E}_{K}(N)\leq N(N-1)I_{K}(\mu),$

and

$\lim_{N\rightarrow\infty}\frac{\mathcal{E}_{K}(N)}{N^{2}}=I_{K}(\mu).$

If $\\{\omega_{N}\\}_{N=2}^{\infty}$ is a sequence of optimal $K$-energy
configurations and $\nu$ is a weak∗ limit point of the normalized counting
measures $\nu_{\omega_{N}}$, then $\nu$ is an equilibrium measure of $I_{K}$.

Thus, if the continuous energy is minimized by the uniform measure, we should
expect uniformly distributed discrete minimizers. That the uniform measure
(uniquely) minimizes the energies follows from the strict positive
definiteness of the kernels, which we show below.

######  Definition 2.13.

We call a kernel _conditionally positive definite_ if for all
$\nu\in\mathcal{Z}(\Omega)$, for which the energy is well defined,
$I_{K}(\nu)\geq 0$. We call a kernel _positive definite_ if for all
$\nu\in\mathcal{M}(\Omega)$ for which the energy is well defined,
$I_{K}(\nu)\geq 0$. We call a kernel _strictly positive definite_ or
_conditionally strictly positive definite_ if it is positive definite or
conditionally positive definite, respectively, and $I_{K}(\nu)=0$ only if
$\nu=0$.

######  Theorem 2.14 (see [16, Theorem 4.2.7]).

If $K$ is conditionally strictly positive definite, then $I_{K}$ has a unique
minimizer in $\mathbb{P}(\Omega)$.

We note that the proofs for Theorems 2.12 and 2.14 given in [16] are for
compact subsets of $\mathbb{R}^{d}$, but they can be generalized to compact
metric spaces.

######  Corollary 2.15.

If $K$ is conditionally strictly positive definite and isometry invariant,
then $\sigma$ is the unique minimizer of $I_{K}$.

###### Proof.

Suppose that $\mu$ minimizes $I_{K}$. Since $K$ is isometry invariant, we have
for all $g\in G$,

$\displaystyle I_{K}(\mu)$
$\displaystyle=\iint\limits_{\Omega\times\Omega}K(x,y)\,d\mu(x)\,d\mu(y)$
$\displaystyle=\iint\limits_{\Omega\times\Omega}K(g(x),g(y))\,d\mu(x)\,d\mu(y)$
$\displaystyle=I_{K}(g_{\\#}(\mu))$

where $g_{\\#}(\mu)$ is the pushforward measure of $\mu$ under $g$. Since
$\mu$ must be unique, it must be isometry invariant, giving us our claim. ∎

######  Lemma 2.16.

If $F\in C([-1,1])$ and $K(x,y)=F(\cos(2\vartheta(x,y)))$ is conditionally
positive definite, then

$F(t)=\sum_{n=0}^{\infty}\widehat{F}(n)P_{n}^{(\alpha,\beta)}(t),$

where the series converges absolutely and uniformly on $[-1,1]$.

###### Proof.

We first observe that $K(x,y)+C$ is positive definite for a large enough
constant $C$ by [15, Theorem 2]. Thus we can assume without loss of generality
that $K(x,y)$ is positive definite. Then the eigenfunctions of the operator
$T_{K}:f\mapsto\int_{\Omega}K(x,y)f(y)\,d\sigma(y)$ are the eigenfunctions of
the Laplace operator and the corresponding eigenvalues are positive. Then the
claim follows from Mercer’s theorem (see, for instance [27]). ∎

######  Theorem 2.17.

If $F\in C([-1,1])$, then $K(x,y)=F(\cos(2\vartheta(x,y)))$ is

  1. (1)

conditionally positive definite if and only if $\widehat{F}(n)\geq 0$ for all
$n\in\mathbb{N}$,

  2. (2)

positive definite if and only if $\widehat{F}(n)\geq 0$ for all
$n\in\mathbb{N}_{0}$,

  3. (3)

conditionally strictly positive definite if and only if $\widehat{F}(n)>0$ for
all $n\in\mathbb{N}$,

  4. (4)

strictly positive definite if and only if $\widehat{F}(n)>0$ for all
$n\in\mathbb{N}_{0}$.

###### Proof.

The forward direction for each of these follows from [15, Theorem 2 and
Section III], with a slight alteration for the strict case.

Part (2) then follows from [15, Lemma 2] and that uniform limits of positive
definite functions are positive definite.

Now, suppose that $\widehat{F}(n)>0$ for all $n\in\mathbb{N}_{0}$. Then $K$ is
positive definite, so by Lemma 2.16, the addition formula (2.7), and the
density of $\\{Y_{n,k}:n\in\mathbb{N}_{0},k\in\\{1,...,\dim(V_{n})\\}\\}$ in
$C(\Omega)$, we have for any $\mu\in\mathcal{M}(\Omega)$, not identically
zero,

$\displaystyle I_{K}(\mu)$
$\displaystyle=\sum_{n=0}^{\infty}\widehat{F}(n)\iint\limits_{\Omega\times\Omega}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))\,d\mu(x)\,d\mu(y)$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=1}^{m_{n}}\frac{\widehat{F}(n)P_{n}^{(\alpha,\beta)}(1)}{m_{n}}\iint\limits_{\Omega\times\Omega}Y_{n,k}(x)Y_{n,k}(y)\,d\mu(x)\,d\mu(y)$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=1}^{m_{n}}\frac{\widehat{F}(n)P_{n}^{(\alpha,\beta)}(1)}{m_{n}}\Big{(}\int_{\Omega}Y_{n,k}(x)\,d\mu(x)\Big{)}^{2}>0,$

proving (4). Parts (1) and (3) now follow from [15, Theorem 2].

∎

######  Theorem 2.18.

For $0\leq s<D$, the Riesz kernel $K_{s}$ is strictly positive definite.

###### Proof.

For $\varepsilon>0$ and $s\geq 0$, let

$K_{s,\varepsilon}(x,y)=\begin{cases}\Big{(}\varepsilon+\frac{1-\cos(2\vartheta(x,y))}{2}\Big{)}^{-\frac{s}{2}}&\text{
for }s>0\\\
-\frac{1}{2}\log\Big{(}\varepsilon+\frac{1-\cos(2\vartheta(x,y))}{2}\Big{)}&\text{
for }s=0\end{cases}$

and

$F_{s,\varepsilon}(t)=\begin{cases}\Big{(}\varepsilon+\frac{1-t}{2}\Big{)}^{-\frac{s}{2}}&\text{
for }s>0\\\ -\frac{1}{2}\log\Big{(}\varepsilon+\frac{1-t}{2}\Big{)}&\text{ for
}s=0\end{cases}.$

Note that for all $x,y\in\Omega$, $K_{s,\varepsilon}(x,y)$ is strictly
decreasing in $\varepsilon$, and $\lim_{\varepsilon\rightarrow
0}K_{s,\varepsilon}(x,y)=K_{s}(x,y)$. Now, suppose that
$\mu\in\mathcal{M}(\Omega)$ (not identically zero) such that $I_{K_{s}}(\mu)$
is well defined. There exists $\mu^{+},\mu^{-}\in\mathcal{B}(\Omega)$ such
that $\mu=\mu^{+}-\mu^{-}$, and so, by the Monotone Convergence Theorem, we
have

$\displaystyle I_{K_{s}}(\mu)$
$\displaystyle=I_{K_{s}}(\mu^{+})-2\iint\limits_{\Omega\times\Omega}K_{s}(x,y)d\mu^{+}(x)d\mu^{-}(y)+I_{K_{s}}(\mu^{-})$
$\displaystyle=\lim_{\varepsilon\rightarrow
0}\left(I_{K_{s,\varepsilon}}(\mu^{+})-2\iint\limits_{\Omega\times\Omega}K_{s,\varepsilon}(x,y)d\mu^{+}(x)d\mu^{-}(y)+I_{K_{s,\varepsilon}}(\mu^{-})\right)$
$\displaystyle=\lim_{\varepsilon\rightarrow 0}I_{K_{s,\varepsilon}}(\mu).$

We now show that $I_{K_{s,\varepsilon}}(\mu)$ is positive and strictly
decreasing as a function of $\varepsilon$, for all $s\geq 0$.

For $\varepsilon>0$,

(2.14)
$\Big{(}\varepsilon+\frac{1-t}{2}\Big{)}^{-\frac{s}{2}}=\Big{(}\varepsilon+1\Big{)}^{-\frac{s}{2}}\sum_{k=0}^{\infty}\binom{k+\frac{s}{2}-1}{k}\Big{(}\frac{t+1}{2(\varepsilon+1)}\Big{)}^{k}$

and

(2.15)
$-\log\Big{(}\varepsilon+\frac{1-t}{2}\Big{)}=-\log\Big{(}\varepsilon+1\Big{)}+\sum_{k=1}^{\infty}\frac{1}{k}\Big{(}\frac{t+1}{2(\varepsilon+1)}\Big{)}^{k},$

with the series converging uniformly on $[-1,1]$.

The polynomials $(t+1)^{k}$ can be expressed as linear combinations of Jacobi
polynomials; the coefficients are given by

$\frac{m_{n}}{P_{n}^{(\alpha,\beta)}(1)^{2}}\int_{0}^{\frac{\pi}{2}}(1+\cos(2\vartheta))^{k}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta))\,d\nu^{\alpha,\beta}(\vartheta)\\\
=\frac{m_{n}}{P_{n}^{(\alpha,\beta)}(1)^{2}}\binom{k}{n}\frac{2^{k}(\alpha+1)_{n}(\beta+1)_{k}}{(\alpha+\beta+2)_{n+k}}.$

Together with the series expansions (2.14) and (2.15) this shows that
$\widehat{F_{s,\varepsilon}}(n)>0$ for all $s\geq 0,n\in\mathbb{N}_{0}$, for
$\varepsilon$ sufficiently small.

These coefficients are positive, meaning that $K_{s,\varepsilon}$ is strictly
positive definite, for $\varepsilon$ sufficiently small, by Theorem 2.17.
Since $\mu\neq 0$, we know (from the density of
$\operatorname{span}(\\{Y_{n,k}:n\in\mathbb{N}_{0},1\leq k\leq\dim(V_{n})\\})$
in $C(\Omega)$) that for some $m\in\mathbb{N}_{0}$, we must have

$\iint\limits_{\Omega\times\Omega}P_{m}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))\,d\mu(x)\,d\mu(y)>0.$

Since the coefficients $\widehat{F_{s,\varepsilon}}(n)$, for
$n\in\mathbb{N}_{0}$, are strictly decreasing in $\varepsilon$ and positive,
the positive definiteness of the Jacobi polynomials and Lemma 2.16 gives us,
for $0\leq s<D$,

$\displaystyle I_{K_{s}}(\mu)$ $\displaystyle=\lim_{\varepsilon\rightarrow
0}I_{K_{s,\varepsilon}}(\mu)$ $\displaystyle=\lim_{\varepsilon\rightarrow
0}\sum_{n=0}^{\infty}\widehat{F_{s,\varepsilon}}(n)\iint\limits_{\Omega\times\Omega}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))\,d\mu(x)\,d\mu(y)$
$\displaystyle\geq\lim_{\varepsilon\rightarrow
0}\widehat{F_{s,\varepsilon}}(m)\iint\limits_{\Omega\times\Omega}P_{m}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))\,d\mu(x)\,d\mu(y)$
$\displaystyle>0.$

This proves conditional strict positive definiteness. ∎

######  Theorem 2.19 (see [7, Proposition 3.14]).

The Green function $G(x,y)$ is conditionally strictly positive definite.

###### Proof of Theorem 2.11.

The first part of our claim follows from Theorems 2.18 and 2.19 and Corollary
2.15. Equations (2.11), (2.12), and (2.13) follow from direct computation or
our assumptions on $G$.

By Theorem 2.12, any weak∗ limit point of $\\{\nu_{\omega_{N}}:N\geq 2\\}$
must be $\sigma$. We know, by the Banach-Alaoglu Theorem, that
$\mathbb{P}(\Omega)$ is weak∗ compact. Thus the sequence $(\nu_{\omega_{N}})$
has a limit point, which has to be $\sigma$ by Theorem 2.12. Thus the sequence
$(\nu_{\omega_{N}})$ converges to $\sigma$ and the second part of our claim
now follows. ∎

###  2.7. The heat kernel

Using the theory of the heat kernel for a compact Riemannian manifold (see,
for example, [48]) we obtain a lower bound for the Green function $G(x,y)$ on
each of the (compact) projective spaces.

The _heat kernel_ on $\Omega$ is the unique function $H_{t}(x,y):=H(t,x,y)\in
C^{\infty}(\mathbb{R}^{+}\times\Omega\times\Omega)$ satisfying

(2.16) $\displaystyle\triangle_{x}H(t,x,y)+\frac{\partial}{\partial
t}H(t,x,y)=0$ (2.17) $\displaystyle\lim_{t\to
0^{+}}\int_{\Omega}H(t,x,y)f(y)\,d\sigma(y)=f(x)$

for each $f\in C^{\infty}(\Omega),$ where $\triangle_{x}=\triangle_{\Omega}$
is the Laplace operator in the variable $x$.

Similar to the case of the Green function, the spectral theorem and the
addition formula (2.7) together imply that $H_{t}(x,y)$ has a series expansion
in terms of the Jacobi polynomials over $L^{2}(\Omega,\sigma)$:

$\begin{split}H_{t}(x,y)=&\sum_{n=0}^{\infty}e^{-4n(n+\alpha+\beta+1)t}\frac{2n+\alpha+\beta+1}{(\alpha+\beta+1)}\\\
&\times\frac{(\alpha+\beta+1)_{n}(\alpha+1)_{n}}{n!(\beta+1)_{n}}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y))).\end{split}$

Contrasting with the formal expansion for $G(x,y)$, the series expansion of
$H_{t}(x,y)$ is uniformly convergent in $x,y\in\Omega$ for all $t>0$.

Integrating $(1-H_{t}(x,y))$ with respect to $t$, we arrive at the kernel

(2.18)
$\begin{split}G_{t}(x,y)=&\sum_{n=1}^{\infty}e^{-4n(n+\alpha+\beta+1)t}\frac{2n+\alpha+\beta+1}{4n(n+\alpha+\beta+1)}\\\
&\times\frac{(\alpha+\beta+2)_{n-1}(\alpha+1)_{n}}{(\beta+1)_{n}n!}P_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y))),\end{split}$

which we may use to provide an explicit lower bound on the Green function.

######  Lemma 2.20.

For all $t>0$ and $x\neq y$ we have

(2.19) $G(x,y)\geq G_{t}(x,y)-t.$

This approximation is originally due to Elkies, and Lang presents a proof in
[41, Lemma 5.2]. We shall also provide a proof, which uses the non-negativity
of the heat kernel.

######  Theorem 2.21.

For all $x,y\in\Omega$ and $t>0,$

$\displaystyle H_{t}(x,y)\geq 0.$

Non-negativity follows from the fact that $\frac{\partial}{\partial
t}+\triangle_{x}$ is a parabolic differential operator, thus satisfying a
strong maximum principle (see [46, Chapter 3, Theorem 3], [41, page 152].)

###### Proof of Lemma 2.20.

We observe that

$G_{t}(x,y)=\int_{\Omega}G(x,z)H_{t}(z,y)\,d\sigma(z).$

This function is defined for all $(x,y)\in\Omega^{2}$ for $t>0$ by the
integrability of $G(x,y)$. Furthermore, for all $x\neq y$ we have

$\displaystyle\triangle_{x}G_{t}(x,y)+\frac{\partial}{\partial t}G_{t}(x,y)$
$\displaystyle=0$ $\displaystyle\lim_{t\to 0}G_{t}(x,y)$
$\displaystyle=G(x,y)$

by (2.16), (2.17), the integrability of $G$, and the continuity of $G$ for
$x\neq y$.

Then, by the defining property of $G$, we have

$\displaystyle\triangle_{x}G_{t}(x,y)=H_{t}(x,y)-1.$

From this, and Theorem 2.21, we derive, for $t_{1}>t_{0}>0$,

$-(t_{1}-t_{0})\leq\int_{t_{0}}^{t_{1}}\left(H_{t}(x,y)-1\right)\,dt=G_{t_{0}}(x,y)-G_{t_{1}}(x,y).$

Taking the limit $t_{0}\to 0$ and using (2.17), we obtain (2.19). ∎

In Section 4.4, we use this result to obtain lower estimates for the Green
energy on each of the projective spaces.

##  3\. Rotation invariant determinantal point processes on two-point
homogeneous spaces

We denote as $\mathscr{X}$ a (simple) random point process in the space
$\Omega$. To describe the process we specify the random variable
$\mathscr{X}(F)$ counting the number of points of the process in $F$, for all
Borel sets $F\subset\Omega$. A process is called _simple_ , if for any
$p\in\Omega$ we have $\mathscr{X}(\\{p\\})\leq 1$, almost surely.

The joint intensities $\rho(x_{1},\ldots,x_{k})$ are functions defined in
$\Omega$ such that for any family of mutually disjoint subsets
$F_{1},\ldots,F_{k}\subset\Omega$

$\mathbb{E}[\mathscr{X}(F_{1})\cdots\mathscr{X}(F_{k})]=\idotsint\limits_{F_{1}\times\cdots\times
F_{k}}\rho(x_{1},\ldots,x_{k})\,d\sigma(x_{1})\cdots\,d\sigma(x_{k}),$

and we assume that $\rho(x_{1},\ldots,x_{k})=0$, when $x_{i}=x_{j}$ for $i\neq
j$.

######  Definition 3.1.

A random point process (see, e.g., [39, Chapter 4]) is called _determinantal_
with kernel $\mathcal{K}:\Omega\times\Omega\rightarrow\mathbb{C}$ if it is
simple and the joint intensities with respect to a background measure $\sigma$
are given by

$\displaystyle\rho(x_{1},\ldots,x_{k})=\det(\mathcal{K}(x_{i},x_{j}))_{1\leq
i,j\leq k},$

for every $k\geq 1$ and $x_{1},\ldots,x_{k}\in\Omega$.

In [39], it is shown that a determinantal process samples exactly $N$ points
if and only if it is associated to the projection of $L^{2}(\Omega,\sigma)$ to
an $N$-dimensional subspace $H$. Let $\phi_{1},\ldots,\phi_{N}$ be an
orthornormal basis of $H$, then the projection kernel is given by

$\mathcal{K}_{H}(x,y)=\sum\limits_{k=1}^{N}\phi_{k}(x)\overline{\phi_{k}(y)}.$

By the Macchi–Soshnikov theorem (see, e.g., [39, Theorem 4.5.5]) the
projection kernel $\mathcal{K}$ defines a determinantal point process.

By Proposition 2.7 the only finite-dimensional $G$-invariant subspaces of
$L^{2}(\Omega,\sigma)$ are finite orthogonal sums of eigenspaces of
$\triangle_{\Omega}$. Thus it is natural to consider the subspace

$H=V_{0}\oplus\cdots\oplus V_{k}$

and the corresponding projection kernel given by

$\displaystyle\mathcal{K}_{n}^{(\alpha,\beta)}(x,y)=\sum_{k=0}^{n}\sum_{m=1}^{m_{k}}Y_{k,m}(x)Y_{k,m}(y),\
\ \ x,y\in\Omega.$

This defines a $G$-invariant determinantal point process.

Using the addition formula (2.7) and (1.2) the kernel
$\mathcal{K}_{n}^{(\alpha,\beta)}(x,y)$ can be written in the form

(3.1)
$\begin{split}\mathcal{K}_{n}^{(\alpha,\beta)}(x,y)=&\sum_{k=0}^{n}\frac{m_{k}}{P_{k}^{(\alpha,\beta)}(1)}P_{k}^{(\alpha,\beta)}(\cos(2\vartheta(x,y))\\\
=&\frac{(\alpha+\beta+2)_{n}}{(\beta+1)_{n}}P_{n}^{(\alpha+1,\beta)}(\cos(2\vartheta(x,y)),\quad
x,y\in\Omega.\end{split}$

Then, for these kernels we have that

(3.2)
$\begin{split}N&=\operatorname{tr}(\mathcal{K}_{n}^{(\alpha,\beta)(x,x)})=\int\limits_{\Omega}\mathcal{K}_{n}^{(\alpha,\beta)}(x,x)\,d\sigma(x)=\mathcal{K}_{n}^{(\alpha,\beta)}(1)\\\
&=\sum_{k=0}^{n}m_{k}=\frac{(\alpha+\beta+2)_{n}(\alpha+2)_{n}}{(\beta+1)_{n}n!}\sim\frac{\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)}\frac{n^{2\alpha+2}}{\Gamma(\alpha+2)},\end{split}$

which by the Macchi–Soshnikov theorem is the number of points sampled by the
determinantal process associated to the kernel
$\mathcal{K}_{n}^{(\alpha,\beta)}$, almost surely. We shall call these
determinantal point processes _harmonic ensembles_.

Determinantal point processes are very convenient probabilistic models for the
study of energy expressions due to the following theorem (see e.g., [39,
Equation (1.2.2)]).

######  Proposition 3.2.

Let $\mathcal{K}(x,y)$ be a projection kernel with trace $N$ in $\Omega$, and
let $\omega_{N}=\\{x_{1},\ldots,x_{N}\\}\subset\Omega$ be $N$ random points
generated by the corresponding determinantal point process $\mathscr{X}$.
Then, for any measurable $f:\Omega\times\Omega\rightarrow[0,\infty)$, we have

$\begin{split}\mathbb{E}_{\mathscr{X}_{N}}&\left(\sum\limits_{k\neq
j}f(x_{k},x_{j})\right)\\\
=&\iint\limits_{\Omega\times\Omega}\left(\mathcal{K}(x,x)\mathcal{K}(y,y)-|\mathcal{K}(x,y)|^{2}\right)f(x,y)\,d\sigma(x)\,d\sigma(y).\end{split}$

##  4\. Bounds for the Green, Riesz, and Logarithmic Energies on Projective
Spaces

One often studies random configurations of points to find upper estimates for
the minimal energy. However, no local repulsion occurs between i.i.d. random
points, meaning that sampled points may concentrate near one another, and so
the expected discrete energy, $N(N-1)I_{K}(\mu)$, is too coarse of an estimate
for a good upper bound. One can prevent this clumping of sampled points by
distributing one point in each part of a partition of $\Omega$ (i.e. jittered
sampling). Alternatively, one can also generate random point sets with local
repulsion built in, using determinantal point processes. Recently,
determinantal point processes have been used to find bounds on energies in
various symmetric spaces (see, e.g. [9, 11, 12, 2, 8, 10, 37, 44]

Here, we use both jittered sampling and the determinantal point processes
introduced in Section 3 to compute the expectations of the discrete Riesz,
Green, and logarithmic energies under these models. These, of course, provide
upper bounds for the minimal energies. For the lower bounds of these energies,
we use linear programming. As in the case of the sphere, the next-order term
in the upper and lower bounds obtained by these ideas have the same orders of
magnitude in terms of the number of points $N$.

###  4.1. Upper bounds using jittered sampling

Since each projective space is a connected Ahlfors regular metric measure
space with finite measure, we may use the following (formulated for our
context):

######  Proposition 4.1 ([30, Theorem 2]).

For each projective space $\Omega$, there exist positive constants $c_{1}$ and
$c_{2}$ such that for all $N$ sufficiently large, there is a partition of
$\Omega$ into $N$ regions each of measure $\frac{1}{N}$, contained in a
geodesic ball of radius $c_{1}N^{-\frac{1}{D}}$, and containing a geodesic
ball of radius $c_{2}N^{-\frac{1}{D}}$.

######  Proposition 4.2.

For the projective space $\Omega$ and $0\leq s<D$, there is some positive
constant $c_{\Omega,s}$ such that for $N\in\mathbb{N}$ sufficiently large,

$\mathcal{E}_{K_{s}}(N)\leq\begin{cases}N^{2}I_{K_{s}}(\sigma)-c_{\Omega,s}N^{1+\frac{s}{D}}&\text{
for }s>0\\\ N^{2}I_{K_{s}}(\sigma)-c_{\Omega,s}N\log(N)&\text{ for
}s=0\end{cases}.$

###### Proof.

Since $\sin(\vartheta)<\vartheta<2\sin(\vartheta)$ for $\vartheta$
sufficiently small, Proposition 4.1 gives us that there exists some positive
constant $c_{3}$ such that for $N$ sufficiently large, there is a partition of
$\Omega$ into $N$ regions, $D_{1}$, …, $D_{N}$, each of measure $\frac{1}{N}$
and contained in a chordal ball of radius $c_{3}N^{-\frac{1}{2\alpha+2}}$.

Letting $d\sigma_{j}(x):=N\mathbf{1}_{D_{j}}d\sigma(x)$, we have for $0<s<D$,

$\displaystyle\mathcal{E}_{K_{s}}(N)\leq\int_{\Omega}\cdots\int_{\Omega}\sum_{i\neq
j}K_{s}(z_{i},z_{j})\,d\sigma_{1}(z_{1})\cdots d\sigma_{N}(z_{N})$
$\displaystyle=N^{2}\sum_{i\neq j}~{}\iint\limits_{D_{i}\times
D_{j}}K_{s}(z_{i},z_{j})\,d\sigma(z_{i})\,d\sigma(z_{j})$
$\displaystyle=N^{2}\Bigg{(}~{}\iint\limits_{\Omega\times\Omega}K_{s}(x,y)\,d\sigma(x)\,d\sigma(y)-\sum_{j=1}^{N}~{}\iint\limits_{D_{j}\times
D_{j}}K_{s}(x,y)\,d\sigma_{j}(x)\,d\sigma_{j}(y)\Bigg{)}$ $\displaystyle\leq
N^{2}I_{K_{s}}(\sigma)-\sum_{j=1}^{N}\frac{1}{\operatorname{diam}(D_{j})^{s}}\leq
N^{2}I_{K_{s}}(\sigma)-\frac{1}{(2c_{3})^{s}}N^{1+\frac{s}{D}}.$

The logarithmic case works similarly. ∎

We immediately have the following corollary

######  Corollary 4.3.

For the projective space $\Omega$ there is some positive constant
$c_{\Omega,G}$ such that for $N\in\mathbb{N}$ sufficiently large,

$\mathcal{E}_{G}(N)\leq\begin{cases}-c_{\Omega,G}N\log(N)&\text{ for
}\Omega=\mathbb{RP}^{2}\\\ -c_{\Omega,G}N^{2-\frac{2}{D}}&\text{ for
}\Omega\neq\mathbb{RP}^{2}\end{cases}.$

###  4.2. Upper bounds using determinantal point processes

In this section we study the expectation of $E_{K_{s}}$ and $E_{G}$ under the
harmonic ensemble given by the projection kernel (3.1). For the Riesz and
logarithmic energies, this amounts to the computation of integrals of the form

$\mathbb{E}_{\mathscr{X}_{N}}\big{[}E_{K_{s}}\big{]}=\iint\limits_{\Omega\times\Omega}\frac{\mathcal{K}_{n}^{(\alpha,\beta)}(1)^{2}-\mathcal{K}_{n}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))^{2}}{\sin(\vartheta(x,y))^{s}}\,d\sigma(x)\,d\sigma(y).$

Using (2.8) this simplifies to

$\frac{(\alpha+\beta+2)_{n}^{2}}{(\beta+1)_{n}^{2}}\int_{0}^{\frac{\pi}{2}}\frac{1}{\sin(\vartheta)^{s}}\left(P_{n}^{(\alpha+1,\beta)}(1)^{2}-P_{n}^{(\alpha+1,\beta)}(\cos(2\vartheta))^{2}\right)\,d\nu^{\alpha,\beta}(\vartheta).$

We first observe that the integral can be written as

$\frac{1}{\gamma^{\alpha,\beta}}\int_{0}^{\frac{\pi}{2}}\frac{P_{n}^{(\alpha+1,\beta)}(1)^{2}-P_{n}^{(\alpha+1,\beta)}(\cos(2\vartheta))^{2}}{\sin(\vartheta)^{2}}\sin(\vartheta)^{2\alpha+3-s}\cos(\vartheta)^{2\beta+1}\,d\vartheta.$

Now the limit of the quotient exists for $\vartheta\to 0$, which shows that
the integral converges if $2\alpha+3-s>-1$. This gives

######  Theorem 4.4.

The expected Riesz $s$-energy
$\mathbb{E}_{\mathscr{X}_{N}}\big{[}E_{K_{s}}\big{]}$ is finite if and only if
$s<D+2$.

Furthermore, using a similar reasoning the integral

(4.1)
$\int_{0}^{\frac{\pi}{2}}\frac{1}{\sin(\vartheta)^{s}}P_{n}^{(\alpha+1,\beta)}(\cos(2\vartheta))^{2}\,d\nu^{\alpha,\beta}(\vartheta)$

converges, if and only if $s<D$. In order to derive the asymptotic behaviour
of these integrals for $n\to\infty$ we use the classical Hilb approximation
for the Jacobi polynomials (see [52, Theorem 8.21.12])

(4.2)
$\begin{split}&\frac{1}{\binom{n+\alpha+1}{n}}P_{n}^{(\alpha+1,\beta)}(\cos(2\vartheta))\sin(\vartheta)^{\alpha+\frac{3}{2}}\cos(\vartheta)^{\beta+\frac{1}{2}}\\\
&=\Gamma(\alpha+2)\tilde{n}^{-\alpha-1}\sqrt{\vartheta}J_{\alpha+1}(2\tilde{n}\vartheta)+\begin{cases}\mathcal{O}\left(\vartheta^{\alpha+3}\right)&\text{for
}0\leq\vartheta\leq\frac{c}{n}\\\
\mathcal{O}\left(\vartheta^{\frac{1}{2}}n^{-\alpha-\frac{5}{2}}\right)&\text{for
}\frac{c}{n}\leq\vartheta\leq\frac{\pi}{4}\end{cases},\end{split}$

where $J_{\alpha}(x)$ denotes the Bessel function and
$\tilde{n}=n+\frac{1}{2}(\alpha+\beta+1)$. The $\mathcal{O}$-terms are uniform
in $\vartheta\in[0,\frac{\pi}{4}]$ (we use a weaker formulation here than what
is known).

For studying the asymptotic behaviour of (4.1) we insert (4.2) and obtain

$\displaystyle\int_{0}^{\frac{\pi}{4}}$
$\displaystyle\frac{1}{\sin(\vartheta)^{s}}P_{n}^{(\alpha+1,\beta)}(\cos(2\vartheta))^{2}\,d\nu^{\alpha,\beta}(\vartheta)$
$\displaystyle=$
$\displaystyle\frac{1}{\gamma^{\alpha,\beta}}{\binom{n+\alpha+1}{n}}^{2}\Gamma(\alpha+2)^{2}\tilde{n}^{-2\alpha-2}\int_{0}^{\frac{\pi}{4}}\frac{\vartheta}{\sin(\vartheta)^{s+2}}J_{\alpha+1}(2\tilde{n}\vartheta)^{2}\,d\vartheta$
$\displaystyle+\mathcal{O}\left(n^{2\alpha+2}\int_{0}^{\frac{1}{n}}\vartheta^{2\alpha+\frac{5}{2}-s}\,d\vartheta+n^{-2}\int_{\frac{1}{n}}^{\frac{\pi}{4}}\vartheta^{-\frac{3}{2}-s}\,d\vartheta\right),$

where we have used the estimates

$J_{\alpha+1}(x)=\begin{cases}\mathcal{O}(x^{\alpha+1})&\text{ for }x\to 0\\\
\mathcal{O}(x^{-\frac{1}{2}})&\text{ for }x\to\infty\end{cases}$

for the first and the second integral in the $\mathcal{O}$-term, respectively.
We also estimated $\sin(\vartheta)$ trivially with $\vartheta$. The error term
then turns into $\mathcal{O}(n^{s-\frac{3}{2}})$.

Thus we are left with the asymptotic evaluation of the integral

$\int_{0}^{\frac{\pi}{4}}\frac{\vartheta}{\sin(\vartheta)^{s+2}}J_{\alpha+1}(2\tilde{n}\vartheta)^{2}\,d\vartheta.$

We substitute $2\tilde{n}\vartheta=\tau$ and split the integral to obtain

$(2\tilde{n})^{s}\Biggl{(}\int_{0}^{\sqrt{n}}\tau^{-s-1}J_{\alpha+1}(\tau)^{2}\,d\tau+\mathcal{O}\left(\frac{1}{n}\right)\\\
+\int_{\sqrt{n}}^{\frac{\tilde{n}\pi}{2}}\frac{\tau}{(2\tilde{n})^{s+2}\sin(\tau/2\tilde{n})^{s+2}}J_{\alpha+1}(\tau)^{2}\,d\tau\Biggr{)};$

here we have replaced $\sin(\vartheta)$ with $\vartheta$ and controlled the
error in the range $\vartheta<1/\sqrt{n}$. The second integral can be
estimated by $\mathcal{O}(n^{-\frac{s+1}{2}})$. For the first integral we use
[43, Section 3.8.5]

(4.3)
$\int_{0}^{\infty}\tau^{-s-1}J_{\alpha+1}(\tau)^{2}\,d\tau=\frac{1}{2^{s+1}}\frac{\Gamma(s+1)\Gamma(\alpha+1-\frac{s}{2})}{\Gamma(\frac{s+1}{2})^{2}\Gamma(\alpha+2+\frac{s}{2})}$

with an error

$\int_{\sqrt{n}}^{\infty}\tau^{-s-1}J_{\alpha+1}(\tau)^{2}\,d\tau=\mathcal{O}(n^{-\frac{s+1}{2}}).$

For the remaining integral we estimate

$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{1}{\sin(\vartheta)^{s}}P_{n}^{(\alpha+1,\beta)}(\cos(2\vartheta))^{2}\,d\nu^{\alpha,\beta}(\vartheta)\\\
=\mathcal{O}\left(\int_{0}^{\frac{\pi}{2}}P_{n}^{(\alpha+1,\beta)}(\cos(2\vartheta))^{2}\,d\nu^{\alpha+1,\beta}(\vartheta)\right)=\mathcal{O}\left(\frac{1}{n}\right).$

Putting everything together, we obtain

$\int_{0}^{\frac{\pi}{2}}\frac{1}{\sin(\vartheta)^{s}}P_{n}^{(\alpha+1,\beta)}(\cos(2\vartheta))^{2}\,d\nu^{\alpha,\beta}(\vartheta)\\\
=\frac{{\binom{n+\alpha+1}{n}}^{2}}{2\gamma_{\alpha,\beta}}\\!\\!\\!~{}~{}\frac{\Gamma(s+1)\Gamma(\alpha+1-\frac{s}{2})}{\Gamma(\frac{s+2}{2})^{2}\Gamma(\alpha+2+\frac{s}{2})}\tilde{n}^{s-2\alpha-2}\\!+\mathcal{O}(n^{\frac{s-1}{2}}).$

This gives

(4.4)
$\begin{split}\mathbb{E}_{\mathscr{X}_{N}}&\big{[}E_{K_{s}}\big{]}=\mathcal{K}_{n}^{(\alpha,\beta)}(1)^{2}\Gamma(\alpha+\beta+2)\Gamma\left(\alpha+1-\frac{s}{2}\right)\\\
&\times\left(\frac{1}{\Gamma(\alpha+\beta+2-\frac{s}{2})}-\frac{\Gamma(s+1)\Gamma(\alpha+2)^{2}}{\Gamma(\frac{s+2}{2})^{2}\Gamma(\alpha+2+\frac{s}{2})\Gamma(\beta+1)}\tilde{n}^{s-2\alpha-2}\right)\\\
&+\mathcal{O}\left(n^{2\alpha+2+\frac{s-1}{2}}\right).\end{split}$

######  Theorem 4.5.

For $0<s<D$ the expected value of the Riesz energy satisfies

$\begin{split}\mathbb{E}_{\mathscr{X}_{N}}\big{[}&E_{K_{s}}\big{]}=I_{K_{s}}(\sigma)\mathcal{K}_{n}^{(\alpha,\beta)}(1)^{2}\\\
-&\frac{\Gamma(s+1)\Gamma(\alpha+1-\frac{s}{2})\Gamma(\beta+1)}{\Gamma(\frac{s}{2}+1)^{2}\Gamma(\alpha+2+\frac{s}{2})\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}n^{s+D}+\mathcal{O}(n^{s+D-1}).\end{split}$

In terms of the number of points $N$ this gives

(4.5)
$\begin{split}\mathcal{E}_{K_{s}}(N)\leq&~{}\mathbb{E}_{\mathscr{X}_{N}}\big{[}E_{K_{s}}\big{]}\\\
=&I_{K_{s}}(\sigma)N^{2}-\frac{\Gamma(s+1)\Gamma(\alpha+1-\frac{s}{2})\Gamma(\beta+1)}{\Gamma(\frac{s}{2}+1)^{2}\Gamma(\alpha+2+\frac{s}{2})\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}\\\
&\times\left(\frac{\Gamma(\alpha+\beta+2)\Gamma(\alpha+2)}{\Gamma(\beta+1)}\right)^{\frac{2\alpha+s+2}{2\alpha+2}}N^{1+\frac{s}{D}}+\mathcal{O}(N^{1+\frac{s-1}{D}}).\end{split}$

For the limiting case $s=D$, we observe that the implicit constant in the
error term in (4.4) remains bounded for $s\to D$. Thus we can take the limit
$s\to D$ to obtain

######  Theorem 4.6.

The expected energy in the limiting case $s=D$ satisfies

$\begin{split}\mathbb{E}_{\mathscr{X}_{N}}\big{[}E_{K_{D}}\big{]}=&\frac{\Gamma(\alpha+\beta+2)}{\Gamma(\beta+1)}\mathcal{K}_{n}^{(\alpha,\beta)}(1)^{2}\Bigl{(}2\log(n)\\\
&+\psi(2\alpha+3)-2\psi(\alpha+2)-\psi(\beta+1)\Bigr{)}+\mathcal{O}(n^{D-1}).\end{split}$

In terms of the number of points $N$ this gives

(4.6)
$\begin{split}&\mathcal{E}_{K_{D}}(N)\leq\mathbb{E}_{\mathscr{X}_{N}}\big{[}E_{K_{D}}\big{]}=\frac{\Gamma(\alpha+\beta+2)}{(\alpha+1)\Gamma(\beta+1)}\\\
&\quad\times
N^{2}\Bigl{(}\log(N)+\log\left(\frac{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}{\Gamma(\beta+2)}\right)\\\
&\quad+2(\alpha+1)\left(\psi(2\alpha+3)-2\psi(\alpha+2)-\psi(\beta+1)\right)\Bigr{)}+\mathcal{O}(N^{2-\frac{1}{D}}).\end{split}$

For the logarithmic energy we follow the same line of reasoning as above. We
first compute the integral

$\int_{0}^{\frac{\pi}{2}}\log\left(\frac{1}{\sin(\vartheta)}\right)\,d\nu^{\alpha,\beta}(\vartheta)=\frac{1}{2}\left(\psi(\alpha+\beta+2)-\psi(\alpha+1)\right),$

which is done by computing

$\left.\frac{\partial}{\partial
s}\int_{0}^{\frac{\pi}{2}}\sin(\vartheta)^{-s}\,d\nu^{\alpha,\beta}(\vartheta)\right|_{s=0}.$

Then we derive the asymptotic behaviour of

$\int_{0}^{\frac{\pi}{2}}\log\left(\frac{1}{\sin(\vartheta)}\right)P_{n}^{(\alpha+1,\beta)}(\cos(2\vartheta))^{2}\,d\nu^{\alpha,\beta}(\vartheta)$

by using (4.2) again. This gives

$\displaystyle\int_{0}^{\frac{\pi}{4}}\log\left(\frac{1}{\sin(\vartheta)}\right)P_{n}^{(\alpha+1,\beta)}(\cos(2\vartheta))^{2}\,d\nu^{\alpha,\beta}(\vartheta)$
$\displaystyle=$
$\displaystyle\frac{1}{\gamma^{\alpha,\beta}}{\binom{n+\alpha+1}{n}}^{2}\Gamma(\alpha+2)^{2}\tilde{n}^{-2\alpha-2}$
$\displaystyle\times$
$\displaystyle\int_{0}^{\frac{\pi}{4}}\log\left(\frac{1}{\sin(\vartheta)}\right)\frac{\vartheta}{\sin(\vartheta)^{2}}J_{\alpha+1}(2\tilde{n}\vartheta)^{2}\,d\vartheta$
$\displaystyle+$
$\displaystyle\mathcal{O}\left(n^{2\alpha+2}\int_{0}^{\frac{1}{n}}\log(\vartheta)\vartheta^{2\alpha+\frac{5}{2}}\,d\vartheta+n^{-2}\int_{\frac{1}{n}}^{\frac{\pi}{4}}\log(\vartheta)\vartheta^{-\frac{3}{2}}\,d\vartheta\right).$

The error term becomes $\mathcal{O}(n^{-\frac{3}{2}}\log(n))$. The remaining
integral is then treated as above, which gives

$\displaystyle\int_{0}^{\frac{\pi}{4}}\log\left(\frac{1}{\sin(\vartheta)}\right)\frac{\vartheta}{\sin(\vartheta)^{2}}J_{\alpha+1}(2\tilde{n}\vartheta)^{2}\,d\vartheta$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\log\left(\frac{2\tilde{n}}{\tau}\right)J_{\alpha+1}(\tau)^{2}\,\frac{d\tau}{\tau}+\mathcal{O}\left(\frac{\log
n}{n}\right)$ $\displaystyle=$ $\displaystyle\frac{1}{2(\alpha+1)}\log(2n)$
$\displaystyle-$
$\displaystyle\frac{1}{4(\alpha+1)^{2}}\left(2(\alpha+1)(\psi(\alpha+1)+\log(2))+1\right)+\mathcal{O}\left(\frac{\log
n}{n}\right),$

where we have used

$\displaystyle\int_{0}^{\infty}\frac{1}{t}J_{\alpha+1}(t)^{2}\,dt$
$\displaystyle=\frac{1}{2(\alpha+1)}$
$\displaystyle\int_{0}^{\infty}\frac{\log(t)}{t}J_{\alpha+1}(t)^{2}\,dt$
$\displaystyle=\frac{1}{2(\alpha+1)}\left(\psi(\alpha+1)+\log(2)\right)+\frac{1}{4(\alpha+1)^{2}},$

which can be derived from (4.3).

The remaining integral

$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\log\left(\frac{1}{\sin(\vartheta)}\right)P_{n}^{(\alpha+1,\beta)}(\cos(2\vartheta))^{2}\,d\nu^{(\alpha,\beta)}(\vartheta)$

can be estimated by $\mathcal{O}(\frac{1}{n})$ as above. Putting everything
together, this shows:

######  Theorem 4.7.

The expected value of the logarithmic energy satisfies

$\begin{split}\mathbb{E}_{\mathscr{X}_{N}}\big{[}&E_{K_{0}}\big{]}=I_{K_{0}}(\sigma)\mathcal{K}_{n}^{(\alpha,\beta)}(1)^{2}\\\
&-\frac{\Gamma(\beta+1)}{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}n^{D}\left(\log(n)-\frac{\psi(\alpha+1)+\psi(\alpha+2)}{2}\right)\\\
&+\mathcal{O}(n^{D-1}\log n).\end{split}$

In terms of the number of points this gives

(4.7)
$\begin{split}\mathcal{E}_{K_{0}}(N)\leq&~{}\mathbb{E}_{\mathscr{X}_{N}}\big{[}E_{K_{0}}\big{]}\\\
=&I_{K_{0}}(\sigma)N^{2}-\frac{N}{D}\Biggl{(}\log(N)+\log\left(\frac{\Gamma(\alpha+2)\Gamma(\alpha+\beta+1)}{\Gamma(\beta+1)}\right)\\\
&-(\alpha+1)(\psi(\alpha+1)+\psi(\alpha+2))\Biggr{)}+\mathcal{O}(N^{1-\frac{1}{D}}\log
N).\end{split}$

######  Remark 4.8.

A different approach to the asymptotic study of integrals of the form (4.1)
was used in [50] and [18]. There the integral was rewritten as a sum using
connection formulas and the orthogonality relations. The generating functions
of these expressions can then be expressed in terms of hypergeometric
functions; in some special cases there is a closed form expression.

By (2.9) and (2.10) the Green energy is closely related to the Riesz energy
with exponent $s=D-2$ (or the logarithmic energy for $D=2$); the main
difference between the Green energy and this Riesz energy is that the integral
of the Green function vanishes. Thus we have

######  Theorem 4.9.

If $\alpha>0$, the expected value of the Green energy under the harmonic
ensemble is given by

$\mathbb{E}_{\mathscr{X}_{N}}\big{[}E_{G}\big{]}=-\frac{\Gamma(\beta+1)^{2}}{4\alpha(2\alpha+1)\Gamma(\alpha+1)^{2}\Gamma(\alpha+\beta+2)^{2}}n^{2D-2}+\mathcal{O}(n^{2D-3}).$

In terms of the number of points $N$ this gives

(4.8)
$\begin{split}&\mathcal{E}_{G}(N)\leq\mathbb{E}_{\mathscr{X}_{N}}\big{[}E_{G}\big{]}\\\
&=-\frac{(\alpha+1)^{2}}{4\alpha(2\alpha+1)}\left(\frac{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}{\Gamma(\beta+1)}\right)^{\frac{1}{\alpha+1}}N^{2-\frac{2}{D}}+\mathcal{O}(N^{2-\frac{3}{D}}).\end{split}$

If $\Omega=\mathbb{RP}^{2}$, then

(4.9)
$\begin{split}\mathcal{E}_{G}(N)&\leq\mathbb{E}_{\mathscr{X}_{N}}\big{[}E_{G}\big{]}\\\
&=-\frac{N}{2}\Biggl{(}\log(N)+\log\left(\frac{3}{4}\right)-1+2\gamma\Biggr{)}+\mathcal{O}(N^{\frac{1}{2}}\log
N).\end{split}$

###  4.3. Lower Bounds for Riesz Energy

In this section we develop lower bounds for the Riesz and logarithmic energies
using a method which originates from [54] for the case of the sphere. This
method has then been refined in [17] to provide matching asymptotic orders for
the upper and lower bounds; a general version, again for the sphere, is given
in [16, Theorem 6.4.4].

We present a slightly simplified proof for the lower bounds. For this purpose
we need some lemmas.

######  Lemma 4.10.

If $K(x,y)=g(\cos(2\vartheta(x,y)))$ is continuous and positive definite, then

$\mathcal{E}_{K}(N)\geq\widehat{g}(0)N^{2}-g(1)N.$

We recall that a function $f:I\to\mathbb{R}$ is called completely monotone on
an interval $I$ if for all $n\geq 0$

$\forall u\in I:(-1)^{n}f^{(n)}(u)\geq 0.$

######  Lemma 4.11.

Let $f$ be completely monotone on $[0,\infty)$, and $g(1-2u)=f(u)$ for
$u\in[0,1]$. Then the coefficients $\widehat{g}(n)$ as given in (1.4) are all
nonnegative for $n\geq 0$.

Taking $u=\sin(\vartheta(x,y))^{2}$, and using (2.2), we see that $g$ is a
function of $\cos(2\vartheta(x,y))$. Moreover $g(t)$ is absolutely monotonic
on $[-1,1]$ (i.e. all derivatives of $g$ are nonnegative). The proof is then
essentially the same as the proof of [16, Theorem 5.2.14], changing Gegenbauer
polynomials and weights to Jacobi polynomials and weights.

Let $f$ be completely monotone on $(0,\infty)$. From Taylor’s formula with the
integral form of the remainder term we obtain, for $u>0$,

$f(u)=\sum_{k=0}^{n}\frac{\delta^{k}}{k!}(-1)^{k}f^{(k)}(u+\delta)+\frac{(-1)^{n+1}}{n!}\int_{0}^{\delta}t^{n}f^{(n+1)}(u+t)\,dt.$

This observation was the main ingredient in [17]. All summands are positive
and finite for $\delta>0$ and $u\in[0,\infty)$. Furthermore, all summands are
positive definite, taking $u=\sin(\vartheta(x,y))^{2}$ by Lemma 4.11 and
Theorem 2.17.

We apply Lemma 4.10 to the function

$F_{n,\delta}(u)=\sum_{k=0}^{n}\frac{\delta^{k}}{k!}(-1)^{k}f^{(k)}(u+\delta)\leq
f(u)-\frac{(-1)^{n+1}}{n!}\int_{0}^{\delta}t^{n}f^{(n+1)}(u+t)\,dt,$

with the inequality being an equality for $u>0$, to obtain

(4.10)
$\begin{split}E_{K_{f}}(\omega_{N})\geq&N^{2}\Biggl{(}\int_{0}^{\frac{\pi}{2}}f(\sin(\vartheta)^{2})\,d\nu^{(\alpha,\beta)}(\vartheta)\\\
&-\frac{(-1)^{n+1}}{n!}\int_{0}^{\delta}t^{n}\int_{0}^{\frac{\pi}{2}}f^{(n+1)}\left(t+\sin(\vartheta)^{2}\right)\,d\nu^{(\alpha,\beta)}(\vartheta)\,dt\Biggr{)}\\\
&-NF_{n,\delta}(0).\end{split}$

We now apply the above observations to the functions

$f_{s}(u)=\begin{cases}u^{-s/2}&\text{for }s>0\\\
-\frac{1}{2}\log(u)&\text{for }s=0.\end{cases}$

Then the derivatives are given by

$f_{s}^{(k)}(u)=(-1)^{k}c_{s,k}f_{s+2k}(u),$

with

$c_{s,k}=\begin{cases}1&\text{for }k=0\\\
\left(\frac{s}{2}\right)_{k}&\text{for }s>0\quad\text{and }k>0\\\
\frac{1}{2}(k-1)!&\text{for }s=0\quad\text{and }k>0.\end{cases}$

Then (4.10) becomes

$\displaystyle E_{K_{s}}$ $\displaystyle(\omega_{N})$ $\displaystyle\geq$
$\displaystyle~{}N^{2}\left(\int\limits_{0}^{\frac{\pi}{2}}\\!\\!f_{s}(\sin(\vartheta)^{2})\,d\nu^{(\alpha,\beta)}(\vartheta)-\frac{c_{s,n+1}}{n!}\\!\int\limits_{0}^{\delta}\\!t^{n}\\!\\!\int\limits_{0}^{\frac{\pi}{2}}\\!\\!\frac{d\nu^{(\alpha,\beta)}(\vartheta)}{(t+\sin(\vartheta)^{2})^{\frac{s}{2}+n+1}}\,dt\right)$
$\displaystyle-N\left(f_{s}(\delta)+\delta^{-\frac{s}{2}}\sum_{k=1}^{n}\frac{c_{s,k}}{k!}\right).$

The inner integral then computes as

$\int\limits_{0}^{\frac{\pi}{2}}\frac{d\nu^{(\alpha,\beta)}(\vartheta)}{(t+\sin(\vartheta)^{2})^{\frac{s}{2}+n+1}}=(1+t)^{-\frac{s}{2}-n-1}{\sideset{{}_{2}}{{}_{1}}{\operatorname{F}}\\!\left(\genfrac{}{}{0.0pt}{0}{\beta+1,\frac{s}{2}+n+1}{\alpha+\beta+2}\middle|\,\frac{1}{1+t}\right)}.$

From standard transformations for hypergeometric functions (see [43, Section
2.4.1]) we obtain the asymptotic equivalent as $t\to 0$

$(1+t)^{-\frac{s}{2}-n-1}{\sideset{{}_{2}}{{}_{1}}{\operatorname{F}}\\!\left(\genfrac{}{}{0.0pt}{0}{\beta+1,\frac{s}{2}+n+1}{\alpha+\beta+2}\middle|\,\frac{1}{1+t}\right)}\\\
\sim\frac{\Gamma(\alpha+\beta+2)\Gamma(\frac{s}{2}+n-\alpha)}{\Gamma(\beta+1)\Gamma(\frac{s}{2}+n+1)}t^{\alpha-\frac{s}{2}-n}$

valid for $n>\alpha-\frac{s}{2}$. Thus we choose $n$ as the smallest integer
with this property and obtain as $\delta\to 0$

$\int_{0}^{\delta}t^{n}\int\limits_{0}^{\frac{\pi}{2}}\frac{d\nu^{(\alpha,\beta)}(\vartheta)}{(t+\sin(\vartheta)^{2})^{\frac{s}{2}+n+1}}\,dt\sim\frac{\Gamma(\alpha+\beta+2)\Gamma(\frac{s}{2}+n-\alpha)\delta^{\alpha+1-\frac{s}{2}}}{\Gamma(\beta+1)\Gamma(\frac{s}{2}+n+1)(\alpha+1-\frac{s}{2})}.$

Choosing $\delta=N^{-\frac{1}{\alpha+1}}$ gives

######  Theorem 4.12.

Let $0<s<D$, then there is a constant $C_{s,D}>0$ such that

$E_{K_{s}}(\omega_{N})\geq I_{K_{s}}(\sigma)N^{2}-C_{s,D}N^{1+\frac{s}{D}}.$

For $s=0$ there is a constant $C_{0,D}>0$ such that

$E_{K_{0}}(\omega_{N})\geq I_{K_{0}}(\sigma)N^{2}-\frac{1}{D}N\log
N-C_{0,D}N.$

###  4.4. Lower bounds for the Green energy

In this section we compute lower estimates for Green energy on each of the
projective spaces. For $\mathbb{RP}^{2}$ the lower bound follows immediately
from the lower bound on the logarithmic energy.

######  Theorem 4.13.

There exists some constant $C_{G}>0$ such that the Green energy of every point
configuration $\omega_{N}$ on $\mathbb{RP}^{2}$, with $N\geq 2$, satisfies

$E_{G}(\omega_{N})\geq-\frac{1}{2}N\log(N)-C_{G}N.$

###### Proof.

This follows immediately from Theorem 4.12 and the fact that

$G(x,y)=K_{0}(x,y)-I_{K_{0}}(\sigma)$

on $\mathbb{RP}^{2}$. ∎

For the other spaces, we employ a method developed in [41, Chapter VI, § 5],
which makes use of Lemma 2.20.

######  Theorem 4.14.

If $\alpha=1/2$ (i.e. $\Omega=\mathbb{RP}^{3}$) , the Green energy of any
collection of distinct points $\\{x_{1},\ldots,x_{N}\\}\subset\Omega$ is
bounded below by

$E_{G}(\omega_{N})\geq-\frac{3}{4}\pi^{\frac{1}{3}}N^{2-2/3}+\mathcal{O}(N\log(N)).$

For $\alpha>1/2$,

$E_{G}(\omega_{N})\geq-\frac{1+\alpha}{4\alpha}\left(\frac{\Gamma\left(\beta+1\right)}{\Gamma(\alpha+\beta+2)}\right)^{\frac{1}{\alpha+1}}N^{2-\frac{2}{D}}+\mathcal{O}(N^{2-\frac{3}{D}}).$

In order to prove Theorem 4.14, we need the following two lemmas. The proof of
the lemmas are given after the proof of Theorem 4.14.

######  Lemma 4.15.

For $\delta,c,d\geq 0$, and as $t\to 0$

$\displaystyle\sum_{k=1}^{\infty}\left(ck+d\right)^{\delta}e^{-2(ck+d)^{2}t}=\frac{(2t)^{-\frac{\delta+1}{2}}}{2c}\Gamma\left(\frac{\delta+1}{2}\right)+\mathcal{O}(t^{-\delta/2})$

######  Lemma 4.16.

For $d\geq 0$,
$\sum_{k=1}^{\infty}\left(2k+d\right)^{-1}e^{-2(2k+d)^{2}t}=\mathcal{O}(\log(t))$,
as $t\to 0$.

###### Proof of Theorem 4.14.

For some $t>0$ and any collection of distinct points
$\\{x_{1},\ldots,x_{N}\\}\subset\Omega$, we get

$\displaystyle\sum\limits_{j\neq
i}^{N}G(x_{j},x_{i})+N(N-1)2t\geq\sum\limits_{j\neq
i}^{N}G_{2t}(x_{j},x_{i}).$

Moreover from (2.18) it follows

$\displaystyle\sum\limits_{j\neq i}^{N}G_{2t}(x_{j},x_{i})=$
$\displaystyle\sum\limits_{j\neq
i}^{N}\sum_{k=1}^{\infty}\frac{e^{-2\lambda_{k}t}}{\lambda_{k}}\sum_{\ell=1}^{m_{k}}Y_{k,\ell}(x_{j})Y_{k,\ell}(x_{i})$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}\frac{1}{\lambda_{k}}\sum_{\ell=1}^{m_{k}}\left(\left|\sum\limits_{j=1}^{N}e^{-\lambda_{k}t}Y_{k,\ell}(x_{j})\right|^{2}\\!\\!\\!-\\!\\!\sum\limits_{j=1}^{N}e^{-2\lambda_{k}t}|Y_{k,\ell}(x_{j})|^{2}\right)$
$\displaystyle\geq$
$\displaystyle-\sum_{k=1}^{\infty}\frac{1}{\lambda_{k}}\sum_{\ell=1}^{m_{k}}\sum\limits_{j=1}^{N}e^{-2\lambda_{k}t}|Y_{k,\ell}(x_{j})|^{2}$
$\displaystyle=$
$\displaystyle-\sum\limits_{j=1}^{N}G_{2t}(x_{j},x_{j})=-NG_{2t}(x,x).$

For all $t\geq 0$ and $x,y\in\Omega$

$\displaystyle
G_{t}(x,y)=\sum_{k=1}^{\infty}\frac{m_{k}}{\lambda_{k}}\frac{P_{k}^{(\alpha,\beta)}(\cos(2\vartheta(x,y)))}{P_{k}^{(\alpha,\beta)}(1)}e^{-\lambda_{k}t},$

therefore $G_{2t}(x,x)$ is equal to

$\displaystyle\sum_{k=1}^{\infty}$
$\displaystyle\frac{2k+\alpha+\beta+1}{4k(k+\alpha+\beta+1)}\frac{(\alpha+\beta+2)_{k-1}(\alpha+1)_{k}}{(\beta+1)_{k}k!}e^{-8k(k+\alpha+\beta+1)t}$
$\displaystyle=$
$\displaystyle\frac{\gamma_{\alpha,\beta}}{2\Gamma(\alpha+1)^{2}}$
$\displaystyle\times$
$\displaystyle\sum_{k=1}^{\infty}\frac{2k+\alpha+\beta+1}{k(k+\alpha+\beta+1)}\frac{\Gamma(k+\alpha+\beta+1)\Gamma(k+\alpha+1)}{\Gamma(k+1)\Gamma(k+\beta+1)}e^{-8k(k+\alpha+\beta+1)t}.$

Taking into account that

$\displaystyle\frac{2k+\alpha+\beta+1}{k(k+\alpha+\beta+1)}$
$\displaystyle\frac{\Gamma(k+\alpha+\beta+1)\Gamma(k+\alpha+1)}{\Gamma(k+1)\Gamma(k+\beta+1)}$
$\displaystyle=~{}$ $\displaystyle
2\left(k+\frac{\alpha+\beta+1}{2}\right)^{{2\alpha-1}}\left(1+\mathcal{O}\left(\frac{1}{k}\right)\right)$

we obtain for $0<t\ll 1$, $G_{2t}(x,x)$ equals

$\displaystyle\frac{\gamma_{\alpha,\beta}}{\Gamma(\alpha+1)^{2}}\sum_{k=1}^{\infty}\left(k+\frac{\alpha+\beta+1}{2}\right)^{{2\alpha-1}}e^{-8k(k+\alpha+\beta+1)t}\left(1+\mathcal{O}\left(\frac{1}{k}\right)\right)$
$\displaystyle=\frac{\gamma_{\alpha,\beta}e^{2(\alpha+\beta+1)^{2}t}}{2^{2\alpha-1}\Gamma(\alpha+1)^{2}}\sum_{k=1}^{\infty}\frac{e^{-2(2x+\alpha+\beta+1)^{2}t}}{\left(2k+\alpha+\beta+1\right)^{{1-2\alpha}}}\left(1+\mathcal{O}\left(\frac{1}{k}\right)\right).$

Applying Lemma 4.15 for $\alpha>1/2,\delta=2\alpha-1,c=2$ and
$d=\alpha+\beta+1$, we obtain

$\displaystyle G_{2t}(x,x)=$
$\displaystyle\frac{e^{2{(\alpha+\beta+1)^{2}t}}\gamma_{\alpha,\beta}}{\Gamma(\alpha+1)^{2}2^{2\alpha-1}}\left(\frac{\Gamma(\alpha)}{2^{\alpha+2}}t^{-\alpha}+\mathcal{O}(t^{-\alpha+1/2})\right)+\mathcal{O}(t^{-\alpha+1/2})$
$\displaystyle=$
$\displaystyle\frac{\Gamma(\alpha)\gamma_{\alpha,\beta}}{\Gamma(\alpha+1)^{2}2^{3\alpha-1}}\left(1+\mathcal{O}(t)\right)\left(t^{-\alpha}+\mathcal{O}(t^{-\alpha+1/2})\right)$
$\displaystyle=$
$\displaystyle\frac{\Gamma(\beta+1)}{\alpha\Gamma(\alpha+\beta+1)^{2}2^{3\alpha+2}}t^{-\alpha}+\mathcal{O}(t^{-\alpha+1/2}).$

By choosing
$t=\frac{1}{8}\left(\frac{\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)}\right)^{\frac{1}{\alpha+1}}N^{-\frac{1}{\alpha+1}}$
in order to obtain a maximal lower bound and applying (3.2), we get

$E_{G}(\omega_{N})\geq-\frac{1+\alpha}{4\alpha}\left(\frac{\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)}\right)^{\frac{1}{\alpha+1}}N^{2-\frac{1}{\alpha+1}}+\mathcal{O}(N^{2-\frac{3}{2(\alpha+1)}}).$

Applying Lemma 4.16 for $\alpha=1/2,\delta=0,c=2$ and $d=\alpha+\beta+1$, we
obtain

$\displaystyle G_{2t}(x,x)$
$\displaystyle=\frac{\gamma_{\alpha,\beta}e^{2t}}{\Gamma(\alpha+1)^{2}}\sum_{k=1}^{\infty}\frac{e^{-2(2x+\alpha+\beta+1)^{2}t}}{\left(2k+\alpha+\beta+1\right)^{{1-2\alpha}}}\left(1+\mathcal{O}\left(\frac{1}{k}\right)\right)$

Due to Lemma 4.16,

$\sum_{k=1}^{\infty}e^{-2(2x+\alpha+\beta+1)^{2}t}\mathcal{O}\left(\frac{1}{k}\right)=\mathcal{O}(\log(t)).$

Furthermore, applying Lemma 4.15,

$\sum_{k=1}^{\infty}e^{-2(2x+\alpha+\beta+1)^{2}t}=2^{-3/2}t^{-1/2}\sqrt{\pi}+\mathcal{O}(1).$

Hence, we obtain

$G_{2t}(x,x)=\frac{\gamma_{\alpha,\beta}}{\Gamma(\alpha+1)^{2}}2^{3/2}t^{-1/2}\sqrt{\pi}+\mathcal{O}(\log(t))$

By choosing
$t=\frac{1}{8}\left(\frac{\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)}\right)^{\frac{1}{\alpha+1}}N^{-\frac{1}{\alpha+1}}$
with $\alpha=1/2$ and $\beta=-1/2$, we get

$E_{G}(\omega_{N})\geq-\frac{3}{4}\pi^{\frac{1}{3}}N^{2-2/3}+\mathcal{O}(N\log(N)).$

∎

###### Proof of Lemma 4.15.

Since

$\displaystyle\sum_{k=1}^{\infty}\left(ck+d\right)^{\delta}e^{-2(ck+d)^{2}t}-\int_{1}^{\infty}\left(cx+d\right)^{\delta}e^{-2(cx+d)^{2}t}dx$

is lower bounded by
$e^{-2(c+d)^{2}t}(c+d)^{\delta}-e^{-\delta/2}\left(\frac{\delta}{4t}\right)^{\delta/2}$
and upper bounded by $e^{-\delta/2}\left(\frac{\delta}{4t}\right)^{\delta/2}$,
we obtain

$\displaystyle\sum_{k=1}^{\infty}\left(ck+d\right)^{\delta}e^{-2(ck+d)^{2}t}$
$\displaystyle=\int_{1}^{\infty}\left(cx+d\right)^{\delta}e^{-2(cx+d)^{2}t}dx+\mathcal{O}(t^{-\delta/2})$
$\displaystyle=\int_{0}^{\infty}\left(cx+d\right)^{\delta}e^{-2(cx+d)^{2}t}dx+\mathcal{O}(t^{-\delta/2}).$

In the last equation we used the fact that

$0\leq\int_{0}^{1}\left(cx+d\right)^{\delta}e^{-2(cx+d)^{2}t}dx\leq
e^{-\delta/2}\left(\frac{\delta}{4t}\right)^{\delta/2}.$

Substituting $y$ by $(cx+d)\sqrt{2t}$ lead to the following equation

$\displaystyle\sum_{k=1}^{\infty}$
$\displaystyle\left(ck+d\right)^{\delta}e^{-2(ck+d)^{2}t}$
$\displaystyle=\int_{d\sqrt{2t}}^{\infty}y^{\delta}e^{-y^{2}}(2t)^{-\delta/2}dy+\mathcal{O}(t^{-\delta/2})$
$\displaystyle=\frac{1}{c}(2t)^{-\frac{\delta+1}{2}}\left(\int_{0}^{\infty}y^{\delta}e^{-y^{2}}dy-\int_{0}^{d\sqrt{2t}}y^{\delta}e^{-y^{2}}dy\right)+\mathcal{O}(t^{-\delta/2})$
$\displaystyle=\frac{1}{2c}(2t)^{-\frac{\delta+1}{2}}\Gamma\left(\frac{\delta+1}{2}\right)+\mathcal{O}(t^{-\delta/2}).$

∎

###### Proof of Lemma 4.16.

The following sum

$\displaystyle\sum_{k=1}^{\infty}\left(2k+d\right)^{-1}e^{-2(2k+d)^{2}t}-\int_{1}^{\infty}\left(2x+d\right)^{-1}e^{-2(2x+d)^{2}t}dx$

is lower bounded by $0$ and upper bounded by $\frac{1}{2+d}e^{-2(2+d)^{2}t}$.
Therefore, we obtain

$\displaystyle\sum_{k=1}^{\infty}\left(2k+d\right)^{-1}e^{-2(2k+d)^{2}t}=\int_{1}^{\infty}\left(2x+d\right)^{-1}e^{-2(2x+d)^{2}t}dx+\mathcal{O}(1).$

Let $y=(2x+d)\sqrt{2t}$, then

$\displaystyle\int_{1}^{\infty}\frac{e^{-2(2x+d)^{2}t}}{\left(2x+d\right)}dx$
$\displaystyle=\int_{(2+d)\sqrt{2t}}^{\infty}\frac{\sqrt{2t}}{y}e^{-y^{2}}(2\sqrt{2t})^{-1}dy$
$\displaystyle=\frac{1}{2}\int_{(2+d)\sqrt{2t}}^{\infty}\frac{1}{y}e^{-y^{2}}dy$
$\displaystyle=\frac{\Gamma(0,(2+d)^{2}2t)}{4}$
$\displaystyle=\frac{1}{4}\left(-\gamma-\log((2+d)^{2}2t)-\sum_{k=1}^{\infty}\frac{(-(2+d)^{2}2t)^{k}}{k!k}\right).$

∎

######  Acknowledgement.

This research was initiated during the workshop “Minimal energy problems with
Riesz potentials” held at the American Institute of Mathematics in May 2021.
We would like to thank Carlos Beltrán, Dmitriy Bilyk, and Damir Ferizović for
their helpful suggestions, and Carlos for inspiring us to pursue this topic at
the AIM workshop.

## References

  * [1] O. Alehyane, S. Asserda, and F. Z. Assila, _Some applications of projective logarithmic potentials_ , J. Math. Anal. Appl. 506 (2022), no. 1, Paper No. 125526, 25.
  * [2] K. Alishahi and M. Zamani, _The spherical ensemble and uniform distribution of points on the sphere_ , Electron. J. Probab. 20 (2015), no. 23, 27.
  * [3] G. E. Andrews, R. Askey, and R. Roy, _Special Functions_ , Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999.
  * [4] S. Asserda, F. Z. Assila, and A. Zeriahi, _Projective logarithmic potentials_ , Indiana Univ. Math. J. 69 (2020), no. 2, 487–516.
  * [5] F. Z. Assila, _Logarithmic potentials on $\mathbb{P}^{n}$_, Comptes Rendus Mathématique 356 (2018), no. 3, 283–287.
  * [6] J. C. Baez, _The octonions_ , Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, 145–205, Errata: ibid. 42 (2005), no. 2, 213.
  * [7] C. Beltrán, N. Corral, and J.G. Criado del Rey, _Discrete and continuous green energy on compact manifolds_ , J. of Approx. Theory 237 (2019), 160–185.
  * [8] C. Beltrán, A. M. Delgado, L. Fernández, and J. F. Sánchez Lara, _On Gegenbauer Point Processes on the unit interval_ , https://arxiv.org/abs/2110.05918, 2021.
  * [9] C. Beltrán and U. Etayo, _The projective ensemble and distribution of points in odd-dimensional spheres_ , Constr. Approx. 48 (2018), no. 1, 163–182.
  * [10] by same author, _A generalization of the spherical ensemble to even-dimensional spheres_ , J. Math. Anal. Appl. 475 (2019), no. 2, 1073–1092.
  * [11] C. Beltrán and D. Ferizović, _Approximation to uniform distribution in $\rm{SO}(3)$_, Constr. Approx. 52 (2020), no. 2, 283–311.
  * [12] C. Beltrán, J. Marzo, and J. Ortega-Cerdà, _Energy and discrepancy of rotationally invariant determinantal point processes in high dimensional spheres_ , J. Complexity 37 (2016), 76–109.
  * [13] M. Berger, P. Gauduchon, and E. Mazet, _Le spectre d’une variété riemannienne_ , Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971.
  * [14] R. G. Bettiol, E. A. Lauret, and P. Piccione, _Full Laplace spectrum of distance spheres in symmetric spaces of rank one_ , https://arxiv.org/abs/2012.02349, 2020.
  * [15] S. Bochner, _Hilbert Distances and Positive Definite Functions_ , Annals of Math. 37 (1941), no. 3, 647–656.
  * [16] S.V. Borodachov, D.P. Hardin, , and E.B. Saff, _Discrete Energy on Rectifiable Sets_ , Monographs in Mathematics, Springer, 2019.
  * [17] J. S. Brauchart, _About the second term of the asymptotics for optimal Riesz energy on the sphere in the potential-theoretical case_ , Integral Transforms Spec. Funct. 17 (2006), no. 5, 321–328.
  * [18] J. S. Brauchart and P. J. Grabner, _Weighted ${L}^{2}$-norms of Gegenbauer polynomials_, Aequationes Math. (2022), to appear, available at https://arxiv.org/abs/2103.08303.
  * [19] J. S. Brauchart, D. P. Hardin, and E. B. Saff, _The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere_ , Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., vol. 578, Amer. Math. Soc., Providence, RI, 2012, pp. 31–61.
  * [20] R. S. Cahn and J. A. Wolf, _Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one_ , Comment. Math. Helv. 51 (1976), no. 1, 1–21.
  * [21] P. G. Casazza and J. Kovacevic, _Uniform tight frames for signal processing and communication_ , Proceedings of SPIE 4478 (2004).
  * [22] X. Chen, D. P. Hardin, and E. B. Saff, _On the search for tight frames of low coherence_ , J. Fourier Anal. Appl. 27 (2021), no. 1, Paper No. 2, 27.
  * [23] J. H. Conway and N. J. A. Sloane, _Sphere Packings, Lattices and Groups_ , third ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999, With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov.
  * [24] M. Craioveanu, M. Puta, and T. M. Rassias, _Old and New Aspects in Spectral Geometry_ , Mathematics and Its Applications, vol. 534, Springer Science + Business Media Dordrecht, 2001.
  * [25] J. G. Criado del Rey, _On the separation distance of minimal Green energy points on compact Riemannian manifolds_ , https://arxiv.org/abs/1901.00779, 2019.
  * [26] R. Feng and S. Zelditch, _Random riesz energies on compact kähler manifolds_ , Trans. Amer. Math. Soc. 365 (2013), no. 10, 5579–5604.
  * [27] J. C. Ferreira and V. A. Menegatto, _Eigenvalues of integral operators defined by smooth positive definite kernels_ , Integral Equations Operator Theory 64 (2009), no. 1, 61–81.
  * [28] H. Freudenthal, _Zur ebenen Oktavengeometrie_ , Nederl. Akad. Wetensch. Proc. Ser. A 56 (1953), 195–200.
  * [29] R. Gangolli, _Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters_ , Annales de l’I. H. P., section B 3 (1967), no. 2, 121–226.
  * [30] G. Gigante and P. Leopardi, _Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces_ , Discrete Comput. Geom. 57 (2017), no. 2, 419–430.
  * [31] E. L. Grinberg, _Spherical harmonics and integral geometry on projective spaces_ , Trans. Amer. Math. Soc. 279 (1983), no. 1, 187–203.
  * [32] D. P. Hardin and E. B. Saff, _Minimal Riesz energy point configurations for rectifiable $d$-dimensional manifolds_, Adv. Math. 193 (2005), no. 1, 174–204.
  * [33] F. R. Harvey, _Spinors and Calibrations_ , Academic Press, 1990.
  * [34] S. Helgason, _Differential Geometry and Symmetric Spaces_ , Pure and Applied Mathematics, vol. 12, Academic Press, 1962.
  * [35] by same author, _The Radon transform for Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds._ , Acta Mathematica (Stockholm) 113 (1965), 153–180.
  * [36] by same author, _Groups and Geometric Analysis, Integral Geometry, Invariant Differential Operators, and Spherical Functions_ , Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, 2000\.
  * [37] M. Hirao, _Finite frames, frame potentials and determinantal point processes on the sphere_ , Statist. Probab. Lett. 176 (2021), Paper No. 109129, 6.
  * [38] R. B. Holmes and V. I. Paulsen, _Optimal frames for erasures_ , Linear Algebra and its Applications 377 (2004), 31–51.
  * [39] M. Krishnapur, Y. Peres, J. Ben Hough, and B. Virág, _Zeros of Gaussian analytic functions and determinantal point processes_ , University Lecture Series, vol. 51, American Mathematical Society, 2009.
  * [40] N. S. Landkof, _Foundations of modern potential theory_ , Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972, Translated from the Russian by A. P. Doohovskoy.
  * [41] S. Lang, _Introduction to Arakelov Theory_ , Springer, New York, 1988\.
  * [42] V. I. Levenshtein, _Universal bounds for codes and designs_ , Handbook of Coding Theory, Vol. I, II, North-Holland, Amsterdam, 1998, pp. 499–648.
  * [43] W. Magnus, F. Oberhettinger, and R. P. Soni, _Formulas and Theorems for the Special Functions of Mathematical Physics_ , enlarged ed., Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966.
  * [44] J. Marzo and J. Ortega-Cerdà, _Expected Riesz energy of some determinantal processes on flat tori_ , Constr. Approx. 47 (2018), no. 1, 75–88.
  * [45] A. Merda and T. N. Davison, _Flexible codebook design for limited feedback systems via sequential smooth optimization on the grassmannian manifold_ , IEEE Transactions on Signal Processing 62 (2014), no. 5, 1305–1318.
  * [46] M. H. Protter and H. F. Weinberger, _Maximum Principles in Differential Equations_ , Springer-Verlag New York Inc., 1984.
  * [47] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, _Integrals and Series. Vol. 3_ , Gordon and Breach Science Publishers, New York, 1990, More special functions, Translated from the Russian by G. G. Gould.
  * [48] S. Rosenberg, _The Laplacian on a Riemannian Manifold_ , London Mathematical Society Student Texts, vol. 31, Cambridge University Press, 1997\.
  * [49] O. Shatalov, _Isometric embeddings $\ell_{2}^{m}\to\ell_{p}^{n}$ and cubature formulas over classical fields_, Ph.D. thesis, Technion, Israel Institute of Technology, Haifa, 2001.
  * [50] M. M. Skriganov, _Stolarsky’s invariance principle for projective spaces II_ , https://doi.org/10.48550/arXiv.1912.12335, Dec 2019.
  * [51] S. Steinerberger, _A Wasserstein inequality and minimal Green energy on compact manifolds_ , J. Funct. Anal. 281 (2021), no. 5, Paper No. 109076, 21.
  * [52] G. Szegő, _Orthogonal Polynomials_ , fourth ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975.
  * [53] J. Tits, _Sur certaines classes d’espaces homogénes de groupes de Lie_ , Acad. Roy. Belg. Cl. Sci. Mem. Coll. 29 (1955), no. 3, 268 pp.
  * [54] G. Wagner, _On means of distances on the surface of a sphere (lower bounds)_ , Pacific J. Math. 144 (1990), no. 2, 389–398.
  * [55] by same author, _On means of distances on the surface of a sphere. II. Upper bounds_ , Pacific J. Math. 154 (1992), no. 2, 381–396.
  * [56] H. C. Wang, _Two-point homogeneous spaces_ , Annals of Mathematics 55 (1952), 177–191.
  * [57] J. A. Wolf, _Harmonic Analysis on Commutative Spaces_ , Mathematical Surveys and Monographs, vol. 142, American Mathematical Society, Providence, RI, 2007.
  * [58] K. Zeller and W. Beekmann, _Theorie der Limitierungsverfahren_ , Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 15, Springer-Verlag, Berlin-New York, 1970, Zweite, erweiterte und verbesserte Auflage.