# Boosted Kerr black hole in the presence of plasma111Presented at the meeting
Recent Progress in Relativistic Astrophysics, 6-8 May 2019 (Shanghai, China).

Carlos A. Benavides-Gallego<EMAIL_ADDRESS>Center for Field Theory
and Particle Physics and Department of Physics, Fudan University, 200433
Shanghai, China  Ahmadjon Abdujabbarov<EMAIL_ADDRESS>Center for Field
Theory and Particle Physics and Department of Physics, Fudan University,
200433 Shanghai, China  Ulugh Beg Astronomical Institute, Astronomicheskaya
33, Tashkent 100052, Uzbekistan<EMAIL_ADDRESS>
###### Abstract

In this work, we obtain the deflection angle for a boosted Kerr black hole in
the weak field approximation using the optics in a curved spacetime developed
by J. L. Synge in 1960. We study the behavior of light in the presence of
plasma by considering different distributions: uniform plasma, singular
isothermal sphere, non-singular isothermal gas sphere, and plasma in a galaxy
cluster. We found that the dragging of the inertial system along with the
boosted parameter $\Lambda$ affect the value of the deflection angle. As an
application, we studied the magnification for both uniform and singular
isothermal distributions.

##  I Introduction

The detection of gravitational waves from the coalescence of two black holes
showed the formation of a rapidly rotating black hole boosted with linear
velocity LIGO16b ; LIGO16c ; LIGO16d . Furthermore, the possible observation
of the electromagnetic counterpart from black hole merger could provide more
information about angular and linear momentum in such systems Morozova14 ;
Lyutikov11 . In this sense, it is important to include the boost parameter
into the Kerr black hole solution to investigate possibles effects on the
gravitational field.

On the other hand, gravitational lensing has been used as a tool to test
Einstein’s theory of gravity and a lot of work has been done in this field by
the scientific community. Starting from the work of Synge on optics in a
curved spacetime Synge60 , for example, it has been shown that the photon
motion is affected by the presence of plasma Kogan10 ; Tsupko12 ; Morozova13 .
Moreover, the effect of plasma around a compact objects on lensing effects has
been studied in Rogers:2017ofq ; Rogers:2016xcc ; Kichenassamy:1985zz ;
Perlick:2017fio ; Perlick:2015vta and references there in. In the literature,
there is also a lot of work devoted to the so-called black hole shadow
Vries1999 ; Abdujabbarov:2017pfw ; Abdujabbarov:2016hnw ; Abdujabbarov:2015xqa
; Abdujabbarov:2015pqp ; Bambi:2008jg ; Bambi:2010hf ; Li:2013jra and
references there in.

Recently, a solution of Einstein’s vacuum field equations which describes a
boosted Kerr black hole relative to an asymptotic Lorentz frame was obtained
in Ref. Soares17 . This solution opens the possibility to study the weak
gravitational lensing effect around a boosted Kerr black hole in the presence
of plasma. Hence, in the present manuscript, we study the gravitational
lensing effect around a boosted black hole by considering different plasma
distributions. The paper is organized as follow. First, we briefly discuss the
optics in curved spacetimes and describe the procedure to obtain the
deflection angle in the weak field approximation following works of Ref.
Kogan10 ; Morozova13 . Then, we present the boosted Kerr metric in both non-
rotating and slowly rotating cases. Next, following the previous discussion,
we find the deflection angle and study its behavior in the presence of plasma,
both for uniform and non-uniform distributions (singular isothermal sphere
(SIS), non-singular isothermal sphere (NSIS), and the case of a plasma in a
galaxy cluster (PGC).) Finally, as an application, we study the magnification
for uniform and SIS plasma distributions.

Throughout the paper, we use the convention in which Greek indices run from 0
to 3, while Latin indices run from 1 to 3. Moreover, we use geometrized units,
where $c=G=1$. This work was presented as a talk at the meeting: Progress in
Relativistic Astrophysics at Fudan University and it is based on the paper:
”Gravitational lensing for a boosted Kerr black hole in the presence of
plasmas” published by the European physics journal C.

##  II Optics in a curved space-time

The optics in a curved space-time was first developed by Synge in 1960 Synge60
. This approach considers a static spacetime metric which describes a weak
gravitational field in an asymptotically flat spacetime. In this sense, the
metric tensor $g_{\mu\nu}$ can be expressed as

$\begin{array}[]{ccc}g^{\alpha\beta}=\eta^{\alpha\beta}-h^{\alpha\beta}&\text{and}&g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}.\end{array}$
(1)

Here $\eta_{\alpha\beta}$ is the Minkowski spacetime metric tensor and
$h_{\alpha,\beta}$ are small perturbation ($h_{\alpha\beta}\ll 1$). Moreover,
because the metric must be asymptotically flat, the small perturbations
satisfies $h_{\alpha\beta}\rightarrow 0$ for $x^{\alpha}\rightarrow\infty$ and
$h^{\alpha\beta}=h_{\alpha\beta}$.

Our main interest is to study the effect of different plasma distributions on
the gravitational lensing for a boosted Kerr black hole. In this sense, it is
necessary to obtain the photon trajectories in the presence of a gravitational
field and then include the plasma contribution into the equations of motion.
To do so, Synge modified the Fermat’s least action principle for light
propagation by considering a dispersive medium. Hence, using the Hamiltonian
formalism, he was able to show that the variational principle

$\delta\left(\int p_{\alpha}dx^{\alpha}\right)=0,$ (2)

along with the condition

$W(x^{\alpha},p_{\alpha})=\frac{1}{2}\left[g^{\alpha\beta}p_{\alpha}p_{\beta}-(n^{2}-1)\left(p_{0}\sqrt{-g^{00}}\right)^{2}\right]=0$
(3)

leads to the following system of differential equations Synge60

$\begin{array}[]{ccc}\frac{dx^{\alpha}}{d\lambda}=\frac{\partial W}{\partial
p_{\alpha}}&\text{and}&\frac{dp_{\alpha}}{d\lambda}=-\frac{\partial
W}{\partial x^{\alpha}},\end{array}$ (4)

where the affine parameter $\lambda$ changes along the light trajectory. These
equations describe the trajectories of photons in a gravitational field. On
the other hand, it is important to point out that the scalar function
$W(x_{\alpha},p_{\alpha})$ has been defined from the relationship between the
phase velocity222The phase velocity is defined as the minimum value of
$u^{\prime 2}=1+\frac{dx_{\alpha}dx^{\alpha}}{(V_{\beta}dx^{\beta})^{2}},$
where $u^{\prime}$ is the velocity of a fictitious particle riding on the
wavefront relative to a time-like world-line $C$ (intersecting the wave) of an
observer with 4-velocity $V^{\mu}$ (see Synge60 for details). $u$ and the
4-vector of the photon momentum $p^{\alpha}$, which is given by Synge60

$\frac{c^{2}}{u^{2}}=n^{2}=1+\frac{p_{\alpha}p^{\alpha}}{(p^{0}\sqrt{-g_{00}})^{2}}.$
(5)

Now, to include the effect of plasma in the equations of motion for photons,
we can consider a static inhomogeneous plasma with a refraction index $n$ that
depends on the space location $x^{i}$. Mathematically, this refraction index
is given by Kogan10 ; Morozova13

$\begin{array}[]{ccc}n^{2}=1-\frac{\omega_{e}^{2}}{[\omega(x^{i})]^{2}}&\text{and}&\omega^{2}_{e}=\frac{4\pi
e^{2}N(x^{i})}{m}=K_{e}N(x^{i}).\end{array}$ (6)

In the last expression, $e$ and $m$ are the electron charge and mass
respectively, $\omega_{e}$ is the plasma frequency, and $N(x^{i})$ is the
electron concentration in an inhomogeneous plasma. Moreover, the photon
frequency $\omega(x^{i})$ depends on the space coordinates $x^{1}$, $x^{2}$,
$x^{3}$ due to gravitational redshift.

It is known that for a static medium in a static gravitational field, the
photon energy can be expressed as Synge60

$p_{0}\sqrt{-g^{00}}=-\frac{1}{c}\hbar\omega(x^{i}).$ (7)

Therefore, after using Eq. (6) the scalar function $W(x^{\alpha},p_{\alpha})$
reduces to

$W(x^{\alpha},p_{\alpha})=\frac{1}{2}\left[g^{\alpha\beta}p_{\alpha}p_{\beta}+\frac{\omega^{2}_{e}\hbar^{2}}{c^{2}}\right],$
(8)

where $\hbar$ is the Planck’s constant. The scalar function expressed in Eq.
(8) has been used in Refs. Kogan10 ; Morozova13 to find the equations of light
propagation for diagonal and non-diagonal spacetimes.

In contrast to flat spacetime in vacuum, where the trajectories of photons are
straight lines, the presence of an arbitrary medium in curved spacetimes makes
photons move along bent trajectories. However, taking into account only small
deviations, it is possible to use the components of the 4-momentum of the
photon moving in a straight line along the $z-$axis as an approximation. This
components are given by Kogan10 ; Morozova13

$\begin{array}[]{ccc}p^{\alpha}=\left(\frac{\hbar\omega}{c},0,0,\frac{n\hbar\omega}{c}\right)&\text{and}&p_{\alpha}=\left(-\frac{\hbar\omega}{c},0,0,\frac{n\hbar\omega}{c}\right).\end{array}$
(9)

Eqs. (9) are known as the null approximation. It is important to point out
that both $\omega$ and $n$ are evaluated at $\infty$. In this sense, we have
introduced the notation in which Kogan10 ; Morozova13

$\begin{array}[]{ccc}\omega=\omega(\infty)&\text{and}&n=n(\infty).\\\
\end{array}$ (10)

Now, in sections II.1 and II.2, we use the equations of motion to compute the
deflection angle for diagonal and non-diagonal spacetimes. We shall see that
it depends on the small perturbations $h_{ij}$ and the plasma distribution.

###  II.1 Equations of light propagation in a diagonal spacetime

In a diagonal spacetime, the non-zero components of metric tensor
$g_{\alpha\beta}$ are those with $\alpha=\beta$. Hence, in this particular
case, the function $W(x^{\alpha},p_{\alpha})$ takes the form

$W(x^{\alpha},p_{\alpha})=\frac{1}{2}\left[g^{00}p^{2}_{0}+g^{lm}p_{l}p_{m}+\frac{\omega^{2}_{e}\hbar^{2}}{c^{2}}\right],$
(11)

and the equations of motion for photos in the presence of plasma (see Eq. (4))
reduces to

$\displaystyle\frac{dx^{i}}{d\lambda}$ $\displaystyle=g^{ij}p_{j}$ (12)
$\displaystyle\frac{dp_{i}}{d\lambda}$
$\displaystyle=-\frac{1}{2}g^{lm}_{\;\;\;\;,i}p_{l}p_{m}-\frac{1}{2}g^{00}_{\;\;\;\;,i}p^{2}_{0}-\frac{1}{2}\frac{\hbar^{2}}{c^{2}}K_{e}N_{,i}.$

Now, due to the null approximation, the 3-vector in the direction of the
photon’s momentum (first equation in Eq. (12)) can be expressed as

$p_{i}=\frac{n\hbar\omega}{c}e_{i},$ (13)

where $e_{i}=(0,0,1)$. Therefore, the second relation in Eq. (12) takes the
form

$\frac{d}{d\lambda}\left(\frac{n\hbar\omega}{c}e_{i}\right)=-\frac{1}{2}g^{lm}_{\;\;\;\;,i}p_{l}p_{m}-\frac{1}{2}g^{00}_{\;\;\;\;,i}p^{2}_{0}-\frac{1}{2}\frac{\hbar^{2}}{c^{2}}K_{e}N_{,i}.$

The last expression can be written as

$\frac{de_{i}}{dz}=-\frac{1}{2}\frac{c^{2}}{n\hbar^{2}\omega^{2}}\left(g^{00}_{\;\;\;\;,i}(p_{0})^{2}+g^{lm}_{\;\;\;\;,i}p_{l}p_{m}+\frac{\hbar^{2}}{c^{2}}K_{e}N_{,i}\right)\\\
-e_{i}\frac{dn}{dz}.$

According to reference Kogan10 , only the components of $e_{i}$ that are
perpendicular to the initial direction of propagation were taken into account.
This means that the contribution to the deflection of photons is due only to
the change in $e_{1}$ and $e_{2}$. Hence, after using the null approximation
$e_{i}=0$ along with the assumption of a weak gravitational field, the last
equation reduces to

$\frac{de_{i}}{dz}=\frac{1}{2}\left(h_{33,i}+\frac{1}{n^{2}}h_{00,i}-\frac{1}{n^{2}\omega^{2}}K_{e}N_{,i}\right),$
(14)

for $i=1,2$. Equation (14) can be used to obtain the deflection angle, which
is defined by Perlick00 ; Perlick:2004tq ; Schneider92

$\vec{\hat{\alpha}}=\mathbf{e}(+\infty)-\mathbf{e}(-\infty).$ (15)

Thus, after integration, we obtain the following expression for $\hat{\alpha}$

$\hat{\alpha}_{i}=\frac{1}{2}\int^{\infty}_{-\infty}\left(h_{33,i}+\frac{\omega^{2}}{\omega^{2}-\omega^{2}_{e}}h_{00,i}-\frac{K_{e}}{\omega^{2}-\omega^{2}_{e}}N_{,i}\right)dz,$
(16)

for $i=1,2$. Note that $\omega_{e}$ and $n$ are evaluated at infinty. Finally,
in terms of the impact parameter, $\hat{\alpha}_{i}$ can be expressed as

$\hat{\alpha}_{b}=\frac{1}{2}\int^{\infty}_{-\infty}\frac{b}{r}\left(\frac{dh_{33}}{dr}+\frac{1}{1-\omega^{2}_{e}/\omega^{2}}\frac{dh_{00}}{dr}-\frac{K_{e}}{\omega^{2}-\omega^{2}_{e}}\frac{dN}{dr}\right).$
(17)

###  II.2 Equations of light propagation in a non-diagonal spacetime

In a non-diagonal spacetime the components of the metric tensor
$g_{\alpha\beta}$ do not vanish for $\alpha\neq\beta$. In this sense, the
escalr function $W(x^{\alpha},p_{\alpha})$ in a non-diagonal spacetime takes
the form

$W(x^{\alpha},p_{\alpha})=\frac{1}{2}\left[g^{00}p^{2}_{0}+2g^{0l}p_{0}p_{l}+g^{lm}p_{l}p_{m}+\frac{\omega^{2}_{e}\hbar^{2}}{c^{2}}\right].$
(18)

Note that Eq. (18) differs from Eq. (11) only in terms of the form
$g^{0l}p_{0}p_{l}$. Therefore, after substitution, the equations of motion for
photons in the presence of plasma reduces to

$\displaystyle\frac{dx^{i}}{d\lambda}$ $\displaystyle=g^{ij}p_{j}$ (19)
$\displaystyle\frac{dp_{i}}{d\lambda}$
$\displaystyle=-\frac{1}{2}g^{lm}_{\;\;\;\;,i}p_{l}p_{m}-\frac{1}{2}g^{00}_{\;\;\;\;,i}p^{2}_{0}-g^{0l}_{\;\;\;\;,i}p_{0}p_{l}-\frac{1}{2}\frac{\hbar^{2}}{c^{2}}K_{e}N_{,i}.$

Once again, the first equation in Eq. (19) can be expressed as Eq. (13).
Hence, after considering the null and the weak field approximation, the second
equation in Eq. (19) can be expressed as

$\frac{dp_{i}}{dz}=\frac{1}{2}\frac{n\hbar\omega}{c}\left(h_{33,i}+\frac{1}{n^{2}}h_{00,i}+\frac{1}{n}h_{03,i}-\frac{K_{e}N_{,i}}{n^{2}\omega^{2}}\right),$
(20)

from which, after integration, the deflection angle for a non-diagonal
spacetime in the presence of plasma is given by

$\hat{\alpha}_{i}=\frac{1}{2}\int^{\infty}_{-\infty}\bigg{(}h_{33,i}+\frac{\omega^{2}}{\omega^{2}-\omega^{2}_{e}}h_{00,i}+\frac{1}{n}h_{03,i}-\frac{K_{e}N_{,i}}{\omega^{2}-\omega^{2}_{e}}\bigg{)}dz\
.$ (21)

##  III Boosted Kerr black hole in the presence of plasma

The boosted Kerr black hole was obtained by I. D. Soares in 2017 Soares17 .
This spacetime is a solution of Einstein’s vacuum field equations which
describe a boosted black hole relative to an asymptotic Lorentz frame. In the
Kerr-Schild coordinates, the line element is given by

$\displaystyle ds^{2}$
$\displaystyle=-\left(1-\frac{2Mr}{\Sigma}\right)dt^{\prime
2}+\left(1+\frac{2Mr}{\Sigma}\right)dr^{2}+\frac{\Sigma}{\Lambda}d\theta^{2}+\frac{A\sin^{2}(\theta)}{\Lambda^{2}\Sigma}d\phi^{2}$
$\displaystyle-\frac{4Mra\sin^{2}\theta}{\Lambda\Sigma}dt^{\prime}d\phi-\frac{4Mr}{\Sigma}dt^{\prime}dr-\frac{2a\sin^{2}\theta}{\Lambda}\left(1-\frac{2Mr}{\Sigma}\right)drd\phi$

with,

$\begin{array}[]{ccc}\Sigma=r^{2}+a^{2}\left(\frac{\beta+\alpha\cos\theta}{\alpha+\beta\cos\theta}\right)^{2},&\Lambda=(\alpha+\beta\cos\theta)^{2},&A=\Sigma^{2}+a^{2}\left(\Sigma+2Mr\right)\sin^{2}\theta.\end{array}$
(22)

Note that the solution has three parameters: mass, rotation, and boots.
Moreover, $a={J}/{M}$ is the specific angular momentum of the compact object
with total mass $M$, $\alpha=\cosh\gamma$, $\beta=\sinh\gamma$, and $\gamma$
is the usual Lorentz factor which defines the boost velocity $v$ by the
formula $v=\tanh\gamma={\beta}/{\alpha}$. The metric in (III) exactly reduces
to Kerr when $\Lambda=1$ ($v=0$). Moreover, it is important to point out that
the direction of the boost for the Kerr black hole is along the axis of
rotation.

In this section, we compute the deflection angle for a boosted Kerr black hole
in the presence of plasma using the weak field approximation discussed in
Sec.II. First, we consider the non-rotating case ($a=0$) which correspond to
the boosted Schwarzschild black hole, and then the slowly rotating case.

###  III.1 Non-rotating case

To study the behavior of $\hat{\alpha}$ in the presence of plasma for the non-
rotating case, it is necessary to express the line element Eq. (III) in
Cartesian coordinates and find the small perturbations $h_{ij}$. Nevertheless,
before doing so, we first consider the limit $v<<1$. Hence, under this
approximation, Eq. (III) takes the form Benavides-Gallego:2018ufb

$\displaystyle ds^{2}$ $\displaystyle=-\left(1-\frac{2M}{r}\right)dt^{\prime
2}+\left(1+\frac{2M}{r}\right)dr^{2}+r^{2}(1-2v\cos\theta)d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}$ $\displaystyle-4vr^{2}\sin^{2}\theta\cos\theta
d\phi^{2}-\frac{4M}{r}dt^{\prime}dr.$

Now, using the coordinate transformation

$\begin{array}[]{cccc}\overline{t}=t,&\overline{x}=r\sin\theta\cos\phi,&\overline{y}=r\sin\theta\sin\phi,&\overline{z}=r\cos\theta.\end{array}$
(23)

Eq. (III.1) reduces to

$\displaystyle ds^{2}$
$\displaystyle=ds^{2}_{0}+h_{11}d\overline{x}^{2}+h_{12}d\overline{x}d\overline{y}+h_{13}d\overline{x}d\overline{z}+h_{22}d\overline{y}^{2}+h_{23}d\overline{y}d\overline{z}+h_{00}dt^{2}+d\overline{z}^{2}h_{33},$
(24)

from which,

$\begin{array}[]{ccc}h_{00}=\frac{2M}{r}&\text{and}&h_{33}=\frac{2M}{r}\cos^{2}\theta-2v\cos\theta\sin^{2}\theta.\end{array}$
(25)

The expressions for $h_{11}$, $h_{12}$, $h_{13}$, $h_{22}$, and $h_{23}$ can
be found in ref. Benavides-Gallego:2018ufb . Here
$ds^{2}_{0}=-dt^{2}+d\overline{x}^{2}+d\overline{y}^{2}+d\overline{z}^{2}$.
Now, after using Eq.(17), the deflection angle in the non-ratating case is
given by

$\hat{\alpha}_{b}=\frac{2M}{b}+\frac{2Mb}{1-\frac{\omega^{2}_{e}}{\omega^{2}}}\int^{\infty}_{0}\frac{dz}{(b^{2}+z^{2})^{\frac{3}{2}}}+\frac{bK_{e}}{2}\int^{\infty}_{-\infty}\frac{1}{\omega^{2}-\omega^{2}_{e}}\frac{1}{r}\frac{dN}{dr}dz.$
(26)

From Eq. (26) we note that, at first order, $\hat{\alpha}_{b}$ does not depend
on the velocity. Hence, if we consider a uniform plasma ($\omega_{e}$
constant), and the approximation $1-n\ll\frac{\omega_{e}}{\omega}$, Eq. (26)
reduces to Kogan10

$\displaystyle\hat{\alpha}_{b}=\frac{2M}{b}\left(1+\frac{1}{1-\frac{\omega^{2}_{e}}{\omega^{2}}}\right).$
(27)

In Fig. 1 left we plotted $\hat{\alpha}_{b}$ as a function of
$\omega^{2}_{e}/\omega^{2}$ for different values of $b/2M$. The plot shows
that $\hat{\alpha}_{b}$ increases as the ration $\omega^{2}_{e}/\omega^{2}$
increases. On the other hand, for small values of $b/2M$ the values of the
deflection angle are greater. For example, for $b/2M=100$ the figure shows
that $\hat{\alpha}_{b}$ is greater than $0.2$; however, for $b/2M=50,100$ the
deflection angle is less than $0.1$. It is also possible to see from the
figure that $\hat{\alpha}_{b}$ has the value $4M/b$ when there is not plasma
($\omega_{e}=0$).

a. b.

Figure 1: a) Plot of $\hat{\alpha}_{b}$ vs. $\omega^{2}_{e}/\omega^{2}$ for
$b/2M=10$ (continuous line), $b/2M=50$ (dashed line), and $b/2M=100$ (dot-
dashed line) for uniform plasma. b) Plot of $\hat{\alpha}_{b}$ vs.
$\omega^{2}_{e}/\omega^{2}$ for the rotating case. We used different values of
the impact parameter: $b/2M=10$ (continuous line), $b/2M=50$ (dashed line),
and $b/2M=100$ (dot-dashed line). We assumed $\Lambda=0.5$,
$J_{r}/M^{2}=0.25$, $\sin\chi=1$, and $\omega^{2}_{e}/\omega^{2}=0.5$. Note
that there is a small increment for $b/2M=10$ when we compare with
Schwarzschild (left panel). Figures taken from Ref. Benavides-Gallego:2018ufb
.

###  III.2 Slowly rotating case

To study the behavior of the deflection angle in the slowly rotating case, we
first express the line element in Eq. (III) in the form Morozova13 ;
Hartle:1968si

$ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})-2\overline{\omega}_{LT}r^{2}\sin^{2}\theta dtd\phi.$ (28)

Figure 2: Schematic representation of the gravitational lensing system. Here,
$\chi$ represents the inclination angle between the vectors $\mathbf{J}_{r}$
and $\mathbf{b}$. In the figure, $D_{s}$, $D_{l}$, and $D_{ls}$ are the
distances from the source to the observer, from the lens to the observer, and
from the source to the lens, respectively. Figures taken from Ref. Benavides-
Gallego:2018ufb .

where $\overline{\omega}_{LT}={2Ma}/{r^{3}}={2\overline{J}}/{r^{3}}$, with
$\overline{J}={J}/{\Lambda}$, is the Lense-Thirring angular velocity of the
dragging of inertial frame. In this sense, due to the presence of non-diagonal
terms in the line element (28), we use the form of $\hat{\alpha}$ obtained in
Eq. (21). Nevertheless, in ordet to obtain the small perturbation $h_{ik}$, we
recall that the dragging effect on the inertial frame contributes to
$\hat{\alpha}$ only by means of the projection $\overline{J}_{r}$ of the
angular momentum Morozova13 . Thus, after the introduction of polar
coordinates $(b,\chi)$ on the intersection point between the light ray and the
$xy$-plane, where $\chi$ is the angle between $\vec{J}_{r}$ and $\vec{b}$, we
find that Morozova13 (see Fig. 2)

$h_{03}=-2\frac{\overline{J}_{r}b\sin\chi}{(b^{2}+z^{2})^{3/2}}.$ (29)

Note that the deflection angle in Eq. (21) will contain two contributions
since $h_{03}$ depends on $b$ and $\chi$333In this manuscript we only consider
de case in which $\chi=\pi/2$ so that $\hat{\alpha}_{\chi}=0$. These
contirbutions are given by Benavides-Gallego:2018ufb

$\displaystyle\hat{\alpha}_{b}$
$\displaystyle=\hat{\alpha}_{bS}-2\overline{J}_{r}\sin\chi\int^{\infty}_{0}\left(\frac{1}{n(b^{2}+z^{2})^{3/2}}-\frac{3b^{2}}{n(b^{2}+z^{2})^{\frac{5}{2}}}\right)dz$
(30) $\displaystyle\hat{\alpha}_{\chi}$
$\displaystyle=-2\overline{J}_{r}\cos\chi\int^{\infty}_{0}\frac{1}{n(b^{2}+z^{2})^{3/2}}dz,$

where $\hat{\alpha}_{bS}$ is the deflection angle for Schwarzschild given by
Eq. (26). Thus, for ${\omega^{2}_{e}}/{\omega^{2}}\ll 1$, $\hat{\alpha}_{b}$
reduces to Benavides-Gallego:2018ufb

$\displaystyle\hat{\alpha}_{b}$
$\displaystyle=\underbrace{\frac{4M}{b}}_{\hat{\alpha}_{S1}}+\underbrace{\frac{2Mb}{\omega^{2}}\int^{\infty}_{0}\frac{\omega^{2}_{e}}{r^{3}}dz}_{\hat{\alpha}_{S2}}+\underbrace{\frac{bK_{e}}{\omega^{2}}\int^{\infty}_{0}\frac{1}{r}\frac{dN}{dr}dz}_{\hat{\alpha}_{S3}}+\underbrace{\frac{bK_{e}}{\omega^{4}}\int^{\infty}_{0}\frac{\omega^{2}_{e}}{r}\frac{dN}{dr}dz}_{\hat{\alpha}_{S4}}$
(31) $\displaystyle+\underbrace{\frac{2J_{r}}{\Lambda
b^{2}}\sin\chi}_{\hat{\alpha}_{B1}}-\underbrace{\frac{J_{r}}{\Lambda\omega^{2}}\sin\chi\int^{\infty}_{0}\frac{\omega_{e}^{2}}{r^{3}}dz}_{\hat{\alpha}_{B2}}+\underbrace{\frac{3b^{2}J_{r}}{\Lambda\omega^{2}}\sin\chi\int^{\infty}_{0}\frac{\omega_{e}^{2}}{r^{5}}dz}_{\hat{\alpha}_{B3}},$

here $r=\sqrt{b^{2}+z^{2}}$, and $S$ and $B$ stand for Schwarzschild and
Boosted, respectively. Equation (31) is similar to that obtained by Kogan et.
al in Ref. Kogan10 : we also find the vacuum gravitational deflection
$\hat{\alpha}_{S1}$, the correction to the gravitational deflection due to the
presence of the plasma $\hat{\alpha}_{S2}$, the refraction deflection due to
the inhomogeneity of the plasma $\hat{\alpha}_{S3}$, and its small correction
$\hat{\alpha}_{S4}$. However, when the boosted Kerr metric is considered,
three more terms appear: $\hat{\alpha}_{B1}$, $\hat{\alpha}_{B2}$, and
$\hat{\alpha}_{B3}$. These are contributions due to the dragging of the
inertial frame. The former is a constant that is presented in all
distributions considered. The other two depend on plasma distribution.

In the presence of uniform plasma, the deflection angle in Eq. (31) takes the
form

$\hat{\alpha}_{b}=\underbrace{\frac{2M}{b}\left(1+\frac{1}{1-\frac{\omega^{2}_{e}}{\omega^{2}}}\right)}_{\hat{\alpha}_{bS}}+\underbrace{\frac{1}{\sqrt{1-\frac{\omega^{2}_{e}}{\omega^{2}}}}\frac{2J_{r}}{b^{2}\Lambda}}_{\hat{\alpha}_{bD}}.$
(32)

In Fig. 3 left panel, we plot $\hat{\alpha}_{bS}$ and $\hat{\alpha}_{b}$ for
the slowly rotating case as a function of the impact parameter $b/2M$. From
the figure, we see that $\hat{\alpha}_{b}$ for a boosted Kerr black hole is
greater than $\hat{\alpha}_{bS}$. This is due to the rotation and boosts
velocity $v$, which is larger for small values of $b/2M$. On the other hand,
for larger values of the impact parameter $b/2M$, this difference becomes very
small, and both angles behave in the same way since
${2J_{r}}/(nb^{2}\Lambda)\rightarrow 0$ when $b/2M\rightarrow\infty$.

a. b.

Figure 3: a) Plot of $\hat{\alpha}_{b}$ vs. $b/2M$ in the presence of uniform
plasma for the slowly rotating (dot-dashed line) and $\hat{\alpha}_{bS}$
(dashed line). In the figure it is also plotted the Schwarzschild case in
vacuum (continuous line). We used $\Lambda=0.5$, $J_{r}/M^{2}=0.25$,
$\sin\chi=1$, and $\omega^{2}_{e}/\omega^{2}=0.5$. b) Plot of
$\hat{\alpha}_{b}$ vs. $\Lambda$ for $J_{r}/M^{2}=0.1$ (continuous line),
$J_{r}/M^{2}=0.2$ (dot-dashed line) , and $J_{r}/M^{2}=0.3$ (dashed line). We
assumed $b/2M=10$, $\sin\chi=1$, and $\omega^{2}_{e}/\omega^{2}=0.5$. Figures
taken from Ref. Benavides-Gallego:2018ufb .

On the other hand, in Fig. 3 right panel, we plot Eq. (32) as a function of
$\Lambda$ for different values of $J_{r}$. We took into account the condition
in which $0<\Lambda\leq 1$ in order to give the values. In this figure, for
different values of $\Lambda$, we see that $\hat{\alpha}_{b}$ is bigger when
$\Lambda\rightarrow 0$. Moreover, for $\Lambda=1$, the deflection angle
reduces to the value $\hat{\alpha}_{bS}+{2J_{r}}/{nb^{2}}$.

Now, in order to study the behavior of $\hat{\alpha}$ in the presence of non-
uniform plasma, it is necessary to know, according to Eq. (32), the plasma
concentration $N(r)$ and plasma frequency $\omega^{2}_{e}$; both functions of
the distribution density $\rho(r)$:

$\begin{array}[]{ccc}N(r)=\frac{\rho(r)}{\kappa
m_{p}}&\text{and}&\omega^{2}_{e}=K_{e}N(r)=\frac{K_{e}\rho(r)}{\kappa
m_{p}},\end{array}$ (33)

where $m_{p}$ is the proton mass and $\kappa$ is is a non-dimensional
coeficient related to the dark matter contribution Kogan10 .

In the case of a singular isothermal sphere (SIS), the distribution density is
given by

$\rho(r)=\frac{\sigma^{2}_{v}}{2\pi r^{2}},$ (34)

where $\sigma^{2}_{v}$ is a one-dimensional velocity dispersion. The SIS model
is often used in lens modeling of galaxies and clusters ChandrasekhaBook39 .
Hence, after using Eq. (31), the deflection angle reduces to Benavides-
Gallego:2018ufb

$\hat{\alpha}_{SIS}=\frac{2}{\overline{b}}+\frac{1}{12\pi}\frac{\omega^{2}_{c}}{\omega^{2}\overline{b}^{3}}-\frac{1}{16}\frac{\omega^{2}_{c}}{\omega^{2}\overline{b}^{2}}+\frac{1}{2}\frac{\widetilde{J}_{r}}{\Lambda\overline{b}^{2}}-\frac{1}{48\pi}\frac{\widetilde{J}_{r}\omega^{2}_{c}}{\Lambda\omega^{2}\overline{b}^{4}}+\frac{1}{20\pi}\frac{\widetilde{J}_{r}\omega^{2}_{c}}{\Lambda\omega^{2}\overline{b}^{4}},$
(35)

where we define $\omega^{2}_{c}=K_{e}\sigma^{2}_{v}/M^{2}\kappa m_{p}$,
$\widetilde{J}_{r}=J_{r}/M^{2}$, and $\overline{b}=b/2M$.

a. b.

Figure 4: a) Plot of $\hat{\alpha}_{SIS}$ vs. $b/2M$ for $\Lambda=1$
(continuous line), $\Lambda=0.2$ (dashed line), and $\Lambda=0.1$ (dot-dashed
line). We used $J_{r}/M^{2}=0.25$, $\sin\chi=1$, and
$\omega^{2}_{c}/\omega^{2}=0.5$. b) Plot of $\hat{\alpha}_{SIS}$ vs. $\Lambda$
for $\widetilde{J}_{r}=0.1$ (continuous line), $\widetilde{J}_{r}=0.2$ (dashed
line), and $\widetilde{J}=0.3$ (dot-dashed line). We used, $\overline{b}=10$,
$\sin\chi=1$, and $\omega^{2}_{c}/\omega^{2}=0.5$. Figures taken from Ref.
Benavides-Gallego:2018ufb

In Fig. 4 left panel, we show the plot of $\hat{\alpha}_{SIS}$ as a function
of $\overline{b}$ for different values of $\Lambda$. According to the figure,
there is no difference for values of $b/2M$ greater than 10. However, for
$b/2M$ near to 10, there is a small difference. This means that
$\hat{\alpha}_{SIS}$ is greater when $\Lambda$ is small. When $\Lambda=1$
($v=0$), we have the case of a slowly rotating massive object. In this sense,
$\Lambda$ has a small effect on the deflection angle. this behavior can be
seen clearly in the right panel of Fig. 4, where it is plotted
$\hat{\alpha}_{SIS}$ as a function of $\Lambda$ for different values of
$\widetilde{J}_{r}$. Note that the boosted parameter is constrained to the
interval $0<\Lambda\leq 1$.

The non-singular isothermal sphere (NSIS) is a model of plasma distribution
where the singularity is replaced by a finite core. In this plasma, the
density distribution is given by Hinshaw87

$\begin{array}[]{ccc}\rho(r)=\frac{\sigma^{2}_{v}}{2\pi(r^{2}+r^{2}_{c})}=\frac{\rho_{0}}{\left(1+\frac{r^{2}}{r^{2}_{c}}\right)}&\text{with}&\rho_{0}=\frac{\sigma^{2}_{v}}{2\pi
r^{2}_{c}}.\end{array}$ (36)

Here $r_{c}$ is the core radius. In the presence of a NSIS, the deflection
angle in Eq. (32) takes the form Benavides-Gallego:2018ufb

$\displaystyle\hat{\alpha}_{NSIS}$
$\displaystyle=\frac{2}{\overline{b}}+\frac{2\overline{b}\omega^{2}_{c}}{\pi\omega^{2}}\bigg{[}\frac{1}{4\overline{b}^{2}\overline{r}^{2}_{c}}-\frac{{\rm
arctanh}\bigg{(}\frac{\overline{r}_{c}}{\sqrt{4\overline{b}^{2}+\overline{r}^{2}_{c}}}\bigg{)}}{\overline{r}^{3}_{c}\sqrt{\overline{r}^{2}_{c}+4\overline{b}^{2}}}\bigg{]}-\frac{1}{2}\frac{\overline{b}\omega^{2}_{c}}{(4\overline{b}^{2}+\overline{r}^{2}_{c})^{\frac{3}{2}}\omega^{2}}+\frac{2\tilde{J}_{r}}{\Lambda\overline{b}^{2}}$
(37)
$\displaystyle-\frac{\widetilde{J}_{r}\omega^{2}_{c}}{2\pi\Lambda\omega^{2}}\left[\frac{1}{4\overline{b}^{2}\overline{r}^{2}_{c}}-\frac{{\rm
arctanh}\left(\frac{\overline{r}_{c}}{\sqrt{4\overline{b}^{2}+\overline{r}^{2}_{c}}}\right)}{\overline{r}^{3}_{c}\sqrt{\overline{r}^{2}_{c}+4\overline{b}^{2}}}\right]+\frac{6}{\pi}\frac{\overline{b}^{2}\widetilde{J}_{r}\omega^{2}_{c}}{\Lambda\omega^{2}}\left[\frac{2\overline{r}^{2}_{c}-12\overline{b}^{2}}{48\overline{b}^{4}\overline{r}^{4}_{c}}+\frac{{\rm
arctanh}\left(\frac{\overline{r}_{c}}{\sqrt{4\overline{b}^{2}+\overline{r}^{2}_{c}}}\right)}{\overline{r}^{5}_{c}\sqrt{\overline{r}^{2}_{c}+4\overline{b}^{2}}}\right].$

In Fig. 5 left panel, we show the behaviour of $\hat{\alpha}_{NSIS}$ as a
function of $\overline{b}$ for different values of $\Lambda$. In the plot,
since we are in the weak field limit, we consider
$\overline{b}\gg\overline{r}_{c}$. The behavior is very similar to the
singular plasma distribution: there are small differences in
$\hat{\alpha}_{NSIS}$ when small values of $\Lambda$ are considered, and no
difference appears when the impact parameter $\overline{b}$ takes values
greater than $10$. Fig. 5 right panel shows clearly this behavior.

a. b.

Figure 5: a) Plot of $\hat{\alpha}_{NSIS}$ vs. $\overline{b}$ for $\Lambda=1$
(continuous line), $\Lambda=0.25$ (dashed line), and $\Lambda=0.1$ (dot-dashed
line). We used, $\widetilde{J}_{r}=0.25$, $\overline{r}_{c}=10$, $\sin\chi=1$,
and $\omega^{2}_{c}/\omega^{2}=0.5$. b) Plot of $\hat{\alpha}_{NSIS}$ vs.
$\Lambda$ for $\widetilde{J}_{r}=0.1$ (continuous line),
$\widetilde{J}_{r}=0.2$ (dashed line), and $\widetilde{J}=0.3$ (dot-dashed
line). We used, $\overline{b}=100$, $\overline{r}_{c}=10$, $\sin\chi=1$, and
$\omega^{2}_{c}/\omega^{2}=0.5$. Note the scale used for the deflection angle:
each value is multiplied by $1e-5=1\times 10^{-5}$. Figures taken from Ref.
Benavides-Gallego:2018ufb .

Finally, we consider the deflection angle in the case of plasma in a galaxy
cluster. Due to the large temperature of electrons in the cluster, the
distribution can be considered as homogeneous. In this sense, it is possible
to suppose a SIS distribution. Hence, the plasma density has the form Kogan10

$\rho(r)=\rho_{0}\left(\frac{r}{r_{0}}\right)^{-s},s=\frac{2\sigma^{2}_{v}}{\mathfrak{R}T}.$
(38)

Therefore, after using Eq. (31), the deflection angle has the form Benavides-
Gallego:2018ufb

$\displaystyle\hat{\alpha}_{PGC}$
$\displaystyle=\frac{2}{\overline{b}}+\frac{\sqrt{\pi}}{2^{s+1}(s+1)}\frac{\overline{r}^{s}_{0}\omega^{2}_{f}}{\overline{b}^{2}\omega^{2}}\frac{\Gamma(\frac{s}{2}+1)}{\Gamma(\frac{s+1}{2})}-\frac{\sqrt{\pi}}{2^{s}}\frac{\omega^{2}_{f}}{\omega^{2}}\frac{\Gamma(\frac{s}{2}+1)}{\Gamma(\frac{s}{2})}\left(\frac{\overline{r}_{0}}{\overline{b}}\right)^{s}+\frac{2\tilde{J}_{r}}{\Lambda\overline{b}^{2}}$
(39)
$\displaystyle-\frac{\pi}{2^{s+2}(s+1)}\frac{\widetilde{J}_{r}\overline{r}^{2}_{0}\omega^{2}_{f}}{\overline{b}^{s+2}\Lambda\omega^{2}}\frac{\Gamma(\frac{s}{2}+1)}{\Gamma(\frac{s+1}{2})}+\frac{3\sqrt{\pi}}{2^{s+2}(s+3)}\frac{\widetilde{J}_{r}\overline{r}^{s}_{0}\omega^{2}_{f}}{b^{s+2}\Lambda\omega^{2}}\frac{\Gamma(\frac{s+4}{2})}{\Gamma(\frac{s+1}{2})}.$

where we define $\omega^{2}_{f}=\frac{K_{e}\rho_{0}}{\kappa m_{p}}$,
$\overline{r}_{0}=r_{0}/M$, $\widetilde{J}_{r}=J_{r}/M^{2}$, and
$\overline{b}=b/2M$.

In Figs. 6 a, b we plot $\hat{\alpha}_{PGC}$ as a function of $\overline{b}$
and $\Lambda$, respectively. In In order to obtain these plots we considered
the case $s<<1$ Kogan10 . According to Figs. 6, differences in the deflection
angle can be seen clearly for the PGC distribution when compared with the
previous distributions. Furthermore, Fig. 6 b shows that the deflection angle
increases due to the dragging and small values of $\Lambda$. In Fig. 6 c, we
show the behavior of $\hat{\alpha}$ as a function of the impact parameter
$\overline{b}$ for all distributions. Observ that values of $\hat{\alpha}$ for
the PGC distribution are grater than the other two distributions. In the
figure, it is also possibel to see the small difference between SIS and NSIS
distributions for small values of $b/2M$.

a. b. c.

Figure 6: a) Plot of $\hat{\alpha}_{PGC}$ vs. $\overline{b}$ for $\Lambda=1$
(continuous line), $\Lambda=0.25$ (dashed line), and $\Lambda=0.1$ (dot-dashed
line). We used, $\widetilde{J}_{r}=0.25$, $\overline{r}_{0}=10$, $\sin\chi=1$,
$s=0.03$, and $\omega^{2}_{f}/\omega^{2}=0.5$. b) Plot of $\hat{\alpha}_{PGC}$
vs. $\Lambda$ for $\widetilde{J}_{r}=0.1$ (continuous line),
$\widetilde{J}_{r}=0.2$ (dashed line), and $\widetilde{J}_{r}=0.3$ (dot-dashed
line). We used $\overline{r}_{0}=10$, $\sin\chi=1$, $s=0.03$,
$\overline{b}=100$ and $\omega^{2}_{f}/\omega^{2}=0.5$. Note the scale used
for the deflection angle: each value is multiplied by $1e-3=1\times 10^{-3}$.
c) Plot of $\hat{\alpha}$ vs. $\overline{b}$ for SIS (continuous line), NSIS
(dashed line), and PGC (dot-dashed line). We used $\Lambda=0.1$,
$\overline{r}_{c}=10$, $\overline{r}_{0}=10$, $\sin\chi=1$, $s=0.03$, and
$\omega^{2}_{f}/\omega^{2}=\omega^{2}_{c}/\omega^{2}=0.5$. For NSIS we use
$\Lambda=1$ since no difference from SIS was found. Figures taken from Ref.
Benavides-Gallego:2018ufb .

##  IV  Lens equation and magnification in the presence of plasma:

In this section, as an application, we study the magnification for boosted
Kerr black hole in the presence of plasma. In particular, we compare the
uniform and singular isothermal distributions.

It is known that the magnification of brightness of the star is defined as
Morozova13

$\begin{array}[]{cc}\mu_{\Sigma}=\frac{I_{tot}}{I_{\ast}}=\sum_{k}\left|\left(\frac{\theta_{k}}{\beta}\right)\left(\frac{d\theta_{k}}{d\beta}\right)\right|,&k=1,2,...,m,\end{array}$
(40)

where $m$ is the number of images, $I_{tot}$ and $I_{\ast}$ are the total and
unlensed brightness of the source respectively, $\theta_{k}$ is the image
position, and $\beta$ is the angular position of the source (see figure 2).
Thus, to compute the magnification for different distributions, it is
necessary to solve the lens equation

$\theta D_{s}=\beta D_{s}+\hat{\alpha}D_{ls}.$ (41)

The lens Eq. (41) relates the distance from the observer to the source $D_{s}$
with the distance from the lens to the source $D_{ls}$. In the last equaiton,
$\hat{\alpha}$ is the deflection angle, and $\theta$, $\beta$ the image and
source positions respectively (see figure 2). In the case of uniform plasma,
Eq. (41) reduces to Benavides-Gallego:2018ufb

$\theta^{3}-\beta\theta^{2}-\frac{\theta^{2}_{E}}{2}\left(1+\frac{1}{1-\frac{\omega^{2}_{e}}{\omega^{2}}}\right)\theta-\frac{\theta^{2}_{E}\tilde{J}_{r}}{4\overline{D}_{l}\Lambda}\frac{1}{\sqrt{1-\frac{\omega^{2}_{e}}{\omega^{2}}}}=0.$
(42)

In the case of the SIS, the lens equation takes the form Benavides-
Gallego:2018ufb

$\theta^{3}-\beta\theta^{2}-\frac{2D_{ls}}{D_{l}D_{s}}\theta-\frac{\overline{D}_{ls}}{\overline{D}^{2}_{l}\overline{D}_{s}}\left(\frac{\tilde{J}_{r}}{2\Lambda}-\frac{\omega^{2}_{c}}{16\omega^{2}}\right)=0.$
(43)

In both cases, we have considered small angles. This means that the impact
parameter can be expressed as $b\approx D_{l}\theta$, with $D_{l}$ denoting
the distance from the observer to the lens.

In Figs. 7.a and 7.c, we plotted the behaviour of the total magnification as a
function of the boosted parameter $\Lambda$ for $\beta=0.001$ and
$\beta=0.0001$ respectively. According to Fig. 7.a, when $\beta=0.001$, the
total magnification decreases as $\Lambda$ increases. This means that
$\mu_{\Sigma tot}$ decreases as the boosted velocity $v$ of the black hole
decreases. A similar behaviour can be seen from Fig. 7.c when $\beta=0.0001$.
Note that for small values of $\beta$, the magnitude of the total
magnification increases. For example: when $\beta=0.001$ the total
magnification is about $\mu_{\Sigma tot}\approx 52.2$. However, when
$\beta=0.0001$, the value increases to $\mu_{\Sigma tot}\approx 522.2$.

a. b. c. d.

Figure 7: a) Plot of $\mu_{\Sigma tot}$ vs. $\Lambda$ when $\beta=0.001$ for
uniform plasma. b) Plot of $\mu_{\Sigma tot}$ vs. $\Lambda$ when $\beta=0.001$
for the SIS distribution. c) Plot of $\mu_{\Sigma tot}$ vs. $\Lambda$ when
$\beta=0.0001$ for unifomr plasma. (d) Plot of $\mu_{\Sigma tot}$ vs.
$\Lambda$ when $\beta=0.0001$ for the SIS distribution. In all the figures we
considered $\overline{D}_{ls}=10$, $\overline{D}_{l}=100$,
$\overline{D}_{s}=110$,
$\omega^{2}_{e}/\omega^{2}=\omega^{2}_{c}/\omega^{2}=0.5$,
$\theta_{E}=0.001818$, and $\tilde{J}_{r}=0.3$. Figures taken from Ref.
Benavides-Gallego:2018ufb .

a. b.

Figure 8: a) Plot of $\mu_{\Sigma tot}$ vs. $\Lambda$ when $\beta=0.001$ for
uniform plasma. b) Plot of $\mu_{\Sigma tot}$ vs. $\Lambda$ when $\beta=0.001$
for the SIS distribution. In all the figures we considered
$\overline{D}_{ls}=10$, $\overline{D}_{l}=100$, $\overline{D}_{s}=110$,
$\omega^{2}_{e}/\omega^{2}=\omega^{2}_{c}/\omega^{2}=0.5$,
$\theta_{E}=0.001818$, and $\tilde{J}_{r}=0.3$. Figures taken from Ref.
Benavides-Gallego:2018ufb .

On the other hand, In Figs. 7.b and 7.d, we plotted the behaviour of
$\mu_{\Sigma tot}$ as a function of the boosted parameter $\Lambda$ for
$\beta=0.001$ and $\beta=0.0001$ respectively. In contrast to uniform plasma,
the total magnification increases as $\Lambda$ increases. Moreover, for small
values of $\beta$, $\mu_{\Sigma tot}$ increases. Finally, in Fig. 8 we show
the behaviour of $\mu_{\Sigma tot}$ as a function of the boosted parameter
$\Lambda$ for uniform plasma and SIS distributions. Note that the behaviour of
$\mu_{\Sigma tot}$ in the SIS distribution is not symmetric.

##  V Conclusions

In this work we studied the deflection angle for a boosted Kerr metric in the
presence of both uniform and non-uniform plasma distributions (where three
different cases were considered). First we studied the deflection angle for
the non-rotating case in the presence of uniform plasma
($\omega_{e}=\text{costant}$) by considering small values of $v$. We found
that $\hat{\alpha}_{b}$ does not dependent, at first order, on the velocity
$v$. It was also found that, after the approximation
$1-n\ll\frac{\omega_{e}}{\omega}$, the deflection angle reduces to the same
expression obtained in Ref. Kogan10 (see Eq. 27). In the case of the slowly
rotating case, the deflection angle $\hat{\alpha}_{b}$ in Eq. (32) contains
two terms: the Schwarzschild angle $\hat{\alpha}_{bS}$, and the contribution
due to the dragging ${\hat{\alpha}_{bD}}$. This result is quite similar to
that of V.S Morozova et al. However, in contrast with their result, Eq. (32)
also depends on the parameter $\Lambda$. Therefore, $\hat{\alpha}$ depends on
$v$ only when the dragging takes place.

In the presence of non-uniform plasma, we consider the deflection angle as a
function of $\overline{b}$ and $\Lambda$ for different distributions. We found
that $\hat{\alpha}$ is affected by the presence of plasma and is greater when
compared with vacuum and uniform distributions. Moreover, we found again that
$\hat{\alpha}$ increases not only due to the dragging, but also when small
values of the boosted parameter $\Lambda$ are considered. In this work, we
also found two important constraints. In the case of NSIS, $r_{c}$ must have
values greater than $6M$. If the core radius $r_{c}$ is smaller than this
limit the deflection angle becomes negative at some point and will not agree
with the usual behavior when $b\rightarrow\infty$ Benavides-Gallego:2018ufb .
On the other hand, regarding the PGC, we found that $s$ must be different from
$-1$ or $-3$ as can be seen from Eq. (39). Nevertheless, this condition is
fulfilled since we consider positive values of $s<<1$.

Finally, we compare the total magnification for uniform and SIS plasma
distributions. According to Fig. 7, for small values of $v$
($0.7\leq\Lambda\leq 1$), the total magnification is greater in the case of
homogeneous plasma. Furthermore, it is important to point out that the total
magnification has small changes in both distributions. In the case of uniform
plasma $\mu_{\Sigma tot}$ ranges from $52.2285$ to $52.2305$, and from
$42.643938$ to $42.643944$ in the SIS. On the other hand, when we compare both
models, we see that the behavior of $\mu_{\Sigma tot}$ is different. When the
boosted Kerr black hole moves towards ($\Lambda>0$) or away ($\Lambda<0$) from
the observer the behavior is very similar (there is a small difference when
$\Lambda\rightarrow-1$ and $\Lambda\rightarrow 1$). However, when we consider
the SIS distribution, the behavior is not symmetric. In both cases, this
behavior is due to cinematic effects.

###### Acknowledgements.

C.A.B.G. acknowledges support from the China Scholarship Council (CSC), grant
No. 2017GXZ019022.

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Strictly Positive Definite Kernels on a Product of Spheres II

Strictly Positive Definite Kernels
on a Product of Spheres II

Jean C. GUELLA, Valdir A. MENEGATTO and Ana P. PERON

J.C. Guella, V.A. Menegatto and A.P. Peron

Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo,
Caixa Postal 668, 13560-970, São Carlos - SP, Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Received June 01, 2016, in final form October 24, 2016; Published online
October 28, 2016

We present, among other things, a necessary and sufficient condition for the
strict positive definiteness of an isotropic and positive definite kernel on
the cartesian product of a circle and a higher dimensional sphere. The result
complements similar results previously obtained for strict positive
definiteness on a product of circles [Positivity, to appear, arXiv:1505.01169]
and on a product of high dimensional spheres [J. Math. Anal. Appl. 435 (2016),
286–301, arXiv:1505.03695].

positive definite kernels; strictly positive definiteness; isotropy;
covariance functions; sphere; circle

33C50; 33C55; 42A16; 42A82; 42C10; 43A35

##  1 Introduction

The theory of positive definite and strictly positive definite kernels on
manifolds and groups can not be separated from the seminal paper of I.J.
Schoenberg [17] published in the first half of the past century. The major
contribution in that paper refers to kernels of the form $(x,y)\in S^{m}\times
S^{m}\to f(x\cdot y)$, in which $\cdot$ is the usual inner product of
$\mathbb{R}^{m+1}$ and the “generating function” $f$ is real and continuous in
$[-1,1]$. The kernel is positive definite if, and only if, the generating
function $f$ has a series representation in the form

$\displaystyle f(t)=\sum_{k=0}^{\infty}a_{k}^{m}P_{k}^{m}(t),\qquad
t\in[-1,1],$

where all the coefficients $a_{k}^{m}$ are nonnegative, $P_{k}^{m}$ denotes
the usual Gegenbauer polynomial of degree $k$ attached to the rational number
$(m-1)/2$ and $\sum_{k}a_{k}^{m}P_{k}^{m}(1)<\infty$. The normalization for
the Gegenbauer polynomials used here is

$\displaystyle P_{n}^{m}(1)=\left(\begin{matrix}n+m-2\\\
n\end{matrix}\right),\qquad n=0,1,\ldots.$

Recall that the positive definiteness of a general real kernel $(x,y)\in
X^{2}\to K(x,y)$ on a nonempty set $X$, demands that $K(x,y)=K(y,x)$, $x,y\in
X$, and the inequality

$\displaystyle\sum_{\mu,\nu=1}^{n}c_{\mu}c_{\nu}K(x_{\mu},x_{\nu})\geq 0,$

whenever $n$ is a positive integer, $x_{1},x_{2},\ldots,x_{n}$ are $n$
distinct points on $X$ and $c_{1},c_{2},\ldots,c_{n}$ are real scalars. For
complex kernels we do not need the symmetry assumption and we have to use
complex scalars instead. Isotropy on $S^{m}$ refers to the fact that the
variables $x$ and $y$ of $S^{m}$ are tied to each other via the inner product
of $\mathbb{R}^{m+1}$.

Some fifty years later, the very same positive definite functions were found
useful for solving scattered data interpolation problems on spheres. But that
demanded strictly positive definite functions, and thus a characterization of
these functions was needed at the start. We recall that the strict positive
definiteness of a general positive definite kernel as in the above definition
requires that the previous inequalities be strict whenever at least one
$c_{\mu}$ is nonzero. In other words, the interpolation matrices
$[K(x_{\mu},x_{\nu})]_{\mu,\nu=1}^{n}$ of $K$ at each set
$\\{x_{1},x_{2},\ldots,x_{n}\\}$ need to be positive definite. The strict
positive definiteness on spheres was an issue for some time until Schoenberg’s
result was complemented by a result of Debao Chen et al. in 2003 [6] and by
Menegatto et al. [14]. A real, continuous, isotropic and positive definite
kernel $(x,y)\in S^{m}\times S^{m}\to f(x\cdot y)$ is strictly positive
definite if, and only if, the following additional condition holds for the
coefficients in Schoenberg’s series representation for the generating function
$f$:

  * –

$(m=1)$ the set $\\{k\in\mathbb{Z}\colon a_{|k|}^{1}>0\\}$ intersects each
full arithmetic progression in $\mathbb{Z}$;

  * –

$(m\geq 2)$ the set $\\{k\colon a_{k}^{m}>0\\}$ contains infinitely many even
and infinitely many odd integers.

As a matter of fact, positive definite and strictly positive definite
functions and kernels on spheres play a fundamental role in several other
applications. For instance, two recent papers authored by Beatson and Zu
Castell [3, 4] provide new families of strictly positive definite functions on
spheres via the so-called half-step operators, a spherical analogue of
Matheron’s montée and descente operators on $\mathbb{R}^{m+1}$. Additional
applications are mentioned in [5, 10].

The spheres belong to a much larger class of metric spaces, that is, they are
compact two-point homogeneous spaces. Positive definiteness on these spaces in
the same sense we explained above was investigated by Gangolli [8] while
strict positive definiteness was completely characterized in [1].

In 2011, the paper [7] considered strictly positive definite functions on
compact abelian groups taking into account a paper on strict positive
definiteness on $S^{1}$ previously written by X. Sun [19]. Among other things,
the paper included abstract characterizations for strict positive definiteness
on a torsion group and on a product of a finite group and a torus.

In the past two years, the attention shifted all the way to positive
definiteness on a product of spaces, the main motivation coming from problems
involving random fields on spaces across time. The first important reference
we would like to mention along this line is [5], where the authors
investigated positive definite kernels on a product of the form $G\times
S^{m}$, in which $G$ is an arbitrary locally compact group, keeping both the
group structure of $G$ and the isotropy of $S^{m}$ in the setting. Let us
denote by $e$ the neutral element of $G$, $*$ the operation of the group $G$
and by $u^{-1}$ the inverse of $u\in G$ with respect to $*$. The main
contribution in [5] states that a continuous kernel of the form
$((u,x),(v,y))\in(G\times S^{m})^{2}\to f(u^{-1}*v,x\cdot y)$ is positive
definite if, and only if, the function $f$ has a representation in the form

$\displaystyle
f(u,t)=\sum_{n=0}^{\infty}f_{n}^{m}(u)P_{n}^{m}(t),\qquad(u,t)\in
G\times[-1,1],$

in which $\\{f_{n}^{m}\\}$ is a sequence of continuous functions on $G$ for
which $\sum f_{n}^{m}(e)P_{n}^{m}(1)<\infty$, with uniform convergence of the
series for $(u,t)\in G\times[-1,1]$. As a matter of fact, the functions
$f_{n}^{m}$ are positive definite on $G$ in the sense that the kernel
$(u,v)\in G^{2}\to f_{n}^{m}(u^{-1}*v)$ is positive definite as previously
defined. The paper [5] did not considered any strict positive definiteness
issues. It is worth to mention that [5] may be linked to [9] where the reader
can find a possible reason for considering positive definiteness on a product
of a locally compact abelian group with a classical space.

Simultaneously, positive definiteness and strict positive definiteness on a
product of spheres was investigated in [11, 12, 13]. A characterization for
the isotropic and positive definite kernels on $S^{m}\times S^{M}$ was deduced
in [12] and that agreed with the characterization mentioned in [5] and also
with [18, Chapter 4]. By the way, a real, continuous and isotropic kernel
$((x,z),(y,w))\in(S^{m}\times S^{M})^{2}\to f(x\cdot y,z\cdot w)$ is positive
definite if, and only if, the function $f$ has a double series representation
in the form

$\displaystyle
f(t,s)=\sum_{k,l=0}^{\infty}a_{k,l}^{m,M}P_{k}^{m}(t)P_{l}^{M}(s),\qquad
t,s\in[-1,1],$ (1.1)

in which $a_{k,l}^{m,M}\geq 0$, $k,l\in\mathbb{Z}_{+}$ and
$\sum\limits_{k,l=0}^{\infty}a_{k,l}^{m,M}P_{k}^{m}(1)P_{l}^{M}(1)<\infty$. As
before, we will call $f$ the generating function of the kernel.

One of the main theorems in [11] reveals that, in the case in which $m,M\geq
2$, a positive definite kernel as above is strictly positive definite if, and
only if, the set $\\{(k,l)\colon a_{k,l}^{m,M}>0\\}$ contains sequences from
each one of the sets $2\mathbb{Z}_{+}\times 2\mathbb{Z}_{+}$,
$2\mathbb{Z}_{+}\times(2\mathbb{Z}_{+}+1)$, $(2\mathbb{Z}_{+}+1)\times
2\mathbb{Z}_{+}$, and $(2\mathbb{Z}_{+}+1)\times(2\mathbb{Z}_{+}+1)$, all of
them having both component sequences unbounded. The very same paper contains
some other intriguing results related to strict positive definiteness,
including a notion of strict positive definiteness that holds in product
spaces only. In the case $m=M=1$, the condition becomes this one [13]: the set
$\\{(k,l)\colon a_{|k|,|l|}^{1,1}>0\\}$ intersects all the translations of
each subgroup of $\mathbb{Z}^{2}$ having the form $\\{(pa,qb)\colon
q,p\in\mathbb{Z}\\}$, $a,b>0$. Even with the completion of these papers, it
became clear very soon that a similar characterization for the remaining case,
that is, strict positive definiteness on $S^{1}\times S^{m}$, $m\geq 2$, would
demand much more work, perhaps a mix of the techniques used in [11, 13].

This is the point where we explain what the contributions in this paper are.
In the next section, we present two abstract results that describe how to
obtain continuous, isotropic and strictly positive kernels on $S^{1}\times
S^{m}$ via the characterization for strict positive definiteness of
continuous, isotropic and positive definite kernels on either $S^{1}$ or
$S^{m}$ separately. In Section 3, we present necessary and sufficient
conditions in order that a real, continuous, isotropic and positive definite
kernel on $S^{1}\times S^{m}$, $m\geq 2$, be strictly positive definite, thus
filling in the missing gap in the previous papers on the subject. In Section
4, we indicate how to obtain a similar characterization after we replace the
sphere $S^{m}$ with an arbitrary compact two-point homogeneous space.

##  2 Abstract sufficient conditions

In this section, we describe two abstract sufficient conditions for strict
positive definiteness on $S^{1}\times S^{m}$, both derived from strict
positive definiteness on single spheres. The results show that transferring
strict positive definiteness from the factors $S^{1}$ and $S^{m}$ to strict
positive definiteness of the product $S^{1}\times S^{m}$ is not so obvious as
it seems.

Here and in the next sections, we will assume all the generating functions of
the kernels are real-valued continuous functions and that the dimension $m$ in
$S^{m}$ is at least $2$. But the reader is advised that the results hold true
for complex kernels after some obvious modifications.

The results to be presented here will depend upon a technical lemma that
provides an alternative formulation for the strict positive definiteness of a
continuous, isotropic and positive definite kernel on $S^{1}\times S^{m}$, to
be described below. Let $(x_{1},w_{1}),(x_{2},w_{2}),\ldots,(x_{n},w_{n})$ be
distinct points on $S^{1}\times S^{m}$ and represent the components in $S^{1}$
in polar form:

$\displaystyle
x_{\mu}=(\cos\theta_{\mu},\sin\theta_{\mu}),\qquad\theta_{\mu}\in[0,2\pi),\qquad\mu=1,2,\ldots,n.$

Write $A$ to denote the interpolation matrix of $((x,z),(y,w))\in(S^{1}\times
S^{m})^{2}\to f(x\cdot y,z\cdot w)$ at
$\\{(x_{1},z_{1}),(x_{2},z_{2}),\ldots,(x_{n},z_{n})\\}$ and consider the
quadratic form $c^{t}Ac$, where $c^{t}$ indicates the transpose of $c$. If the
kernel is positive definite, then the addition theorem for spherical harmonics
[15, p. 18] and the representation (1.1) imply that

$\displaystyle
c^{t}Ac=\sum_{k,l=0}^{\infty}C(k,l,m)a_{k,l}^{1,m}\sum_{j=1}^{d(l,m)}\left|\sum_{\mu=1}^{n}c_{\mu}e^{ik\theta_{\mu}}Y_{l,j}^{m}(z_{\mu})\right|^{2},$

in which $c=(c_{1},c_{2},\ldots,c_{n})$, $\\{Y_{l,j}^{m}\colon
j=1,2,\ldots,d(l,m)\\}$ is a basis for the space of all spherical harmonics of
degree $l$ in $m+1$ variables and $C(k,l,m)$ is a positive constant that
depends upon $k$, $l$ and $m$. The deduction of the equality above requires
the formulas $P_{0}^{1}\equiv 1$ and

$\displaystyle P_{k}^{1}(\cos\theta)=\frac{2}{k}\cos k\theta,\qquad
t\in[0,2\pi),\qquad k=1,2,\ldots.$

The equality $c^{t}Ac=0$ is equivalent to

$\displaystyle\sum_{\mu=1}^{n}c_{\mu}e^{ik\theta_{\mu}}Y_{l,j}^{m}(z_{\mu})=0,\qquad
j=1,2,\ldots,d(l,m),\qquad(k,l)\in\big{\\{}(k,l)\colon
a_{k,l}^{1,m}>0\big{\\}}.$

If this last piece of information holds, we can invoke the addition theorem
once again in order to see that

$\displaystyle\sum_{\mu=1}^{n}c_{\mu}e^{ik\theta_{\mu}}P_{l}^{m}(z_{\mu}\cdot
z)=0,\qquad(k,l)\in\big{\\{}(k,l)\colon a_{k,l}^{1,m}>0\big{\\}},\qquad z\in
S^{m}.$

Finally, if the condition above holds, we can multiply the equality in it by
$e^{-i\theta}$ and split the equation once again via the addition theorem to
obtain

$\displaystyle\sum_{j=1}^{d(l,m)}\left[\sum_{\mu=1}^{n}c_{\mu}e^{ik(\theta_{\mu}-\theta)}Y_{l,j}^{m}(z_{\mu})\right]Y_{l,j}^{m}(z)=0,\qquad(k,l)\in\big{\\{}(k,l)\colon
a_{k,l}^{1,m}>0\big{\\}},\qquad z\in S^{m}.$

Since $\\{Y_{l,j}^{m}\colon j=1,2,\ldots,d(l,m)\\}$ is linearly independent,
we reach

$\displaystyle\sum_{\mu=1}^{n}c_{\mu}Y_{l,j}^{m}(z_{\mu})e^{ik(\theta_{\mu}-\theta)}=0,\qquad
j=1,2,\ldots,d(l,m),\qquad(k,l)\in\big{\\{}(k,l)\colon
a_{k,l}^{1,m}>0\big{\\}},$

for $\theta\in[0,2\pi)$. Multiplying the real part of the equality in the
previous formula by $\cos k\theta$ and integrating in $[0,2\pi]$ with respect
to $\theta$, we obtain

$\displaystyle\sum_{\mu=1}^{n}c_{\mu}Y_{l,j}^{m}(z_{\mu})\cos
k\theta_{\mu}=0,\qquad j=1,2,\ldots,d(l,m),\qquad(k,l)\in\big{\\{}(k,l)\colon
a_{k,l}^{1,m}>0\big{\\}}.$

Similarly, it is easily seen that

$\displaystyle\sum_{\mu=1}^{n}c_{\mu}Y_{l,j}^{m}(z_{\mu})\sin
k\theta_{\mu}=0,\qquad j=1,2,\ldots,d(l,m),\qquad(k,l)\in\big{\\{}(k,l)\colon
a_{k,l}^{1,m}>0\big{\\}}.$

The above computations justify the following lemma.

######  Lemma 2.1.

Let $f$ be the generating function of an isotropic and positive definite
kernel on $S^{1}\times S^{m}$ and consider its series representation (1.1).
The following statements are equivalent:

  1. $(i)$

the kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$
is strictly positive definite;

  2. $(ii)$

if $n$ is a positive integer and
$(x_{1},z_{1}),(x_{2},z_{2}),\ldots,(x_{n},z_{n})$ are distinct points on
$S^{1}\times S^{m}$, then the only solution $c=(c_{1},c_{2},\ldots,c_{n})$ of
the system

$\displaystyle\sum_{\mu=1}^{n}c_{\mu}e^{ik\theta_{\mu}}P_{l}^{m}(z_{\mu}\cdot
z)=0,\qquad(k,l)\in\big{\\{}(k,l)\colon a_{k,l}^{1,m}>0\big{\\}},\qquad z\in
S^{m},$

is $c=0$.

A particular case of the lemma is pertinent (see Theorem 2 in [6]).

######  Lemma 2.2.

Let $g$ be the generating function of an isotropic and positive definite
kernel on $S^{m}$ and consider its series representation according to
Schoenberg. The following statements are equivalent:

  1. $(i)$

the kernel $(z,w)\in(S^{m})^{2}\to g(z\cdot w)$ is strictly positive definite;

  2. $(ii)$

if $n$ is a positive integer and $z_{1},z_{2},\ldots,z_{n}$ are distinct
points on $S^{m}$, then the only solution $c=(c_{1},c_{2},\ldots,c_{n})$ of
the system

$\displaystyle\sum_{\mu=1}^{n}c_{\mu}P_{k}^{m}(z_{\mu}\cdot z)=0,\qquad
k\in\big{\\{}k\colon a_{k}^{m}>0\big{\\}},\qquad z\in S^{m},$

is $c=0$.

Another consequence is this one (see Theorem 5.1 in [16]).

######  Lemma 2.3.

Let $g$ be the generating function of an isotropic and positive definite
kernel on $S^{1}$ and consider its series representation according to
Schoenberg. The following statements are equivalent:

  1. $(i)$

the kernel $(x,y)\in(S^{1})^{2}\to g(x\cdot y)$ is strictly positive definite;

  2. $(ii)$

if $n$ is a positive integer and $x_{1},x_{2},\ldots,x_{n}$ are distinct
points on $S^{1}$ given in polar form
$x_{\mu}=(\cos\theta_{\mu},\sin\theta_{\mu})$, $\mu=1,2,\ldots,n$, then the
only solution $c=(c_{1},c_{2},\ldots,c_{n})$ of the system

$\displaystyle\sum_{\mu=1}^{n}c_{\mu}e^{ik\theta_{\mu}}=0,\qquad
k\in\big{\\{}k\colon a_{k}^{1}>0\big{\\}},$

is $c=0$.

If $f$ generates an isotropic positive definite kernel on $S^{1}\times S^{m}$,
we will adopt the following notation attached to its double series
representation (1.1):

$\displaystyle J_{f}:=\big{\\{}(k,l)\colon a_{k,l}^{1,m}>0\big{\\}}.$

If $I$ is a subset of $\mathbb{Z}_{+}$, we will write $I\in{\rm SPD}(S^{m})$
to indicate that there exists a continuous and positive definite kernel
$(z,w)\in(S^{m})^{2}\to g(z\cdot w)$ for which the set $\\{l\colon
a_{l}^{m}>0\\}$ attached to the series representation for $g$ in Schoenberg’s
result is precisely $I$. This definition is well posed once strict positive
definiteness does not depend upon the actual values of the numbers $a_{l}^{m}$
in the series representation for the generating function but only on the set
$\\{l\colon a_{l}^{m}>0\\}$ itself.

The first important contribution in this section is as follows.

######  Theorem 2.4.

Let $f$ be the generating function of an isotropic and positive definite
kernel on $S^{1}\times S^{m}$ and consider its series representation (1.1). If

$\displaystyle\big{\\{}k\colon\\{l\colon(k,l)\in J_{f}\\}\in{\rm
SPD}\big{(}S^{m}\big{)}\big{\\}}\in{\rm SPD}\big{(}S^{1}\big{)},$

then the kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot
w)$ is strictly positive definite on $S^{1}\times S^{m}$.

###### Proof.

We will show that, under the assumption

$\displaystyle\big{\\{}k\colon\\{l\colon(k,l)\in J_{f}\\}\in{\rm
SPD}\big{(}S^{m}\big{)}\big{\\}}\in{\rm SPD}\big{(}S^{1}\big{)},$

the alternative condition in Lemma 2.1 holds. In particular, the notation
employed in that lemma will be adopted here. Let
$(x_{1},z_{1}),(x_{2},z_{2}),\ldots,$ $(x_{n},z_{n})$ be distinct points in
$S^{1}\times S^{m}$ and suppose that

$\displaystyle\sum_{\mu=1}^{n}c_{\mu}e^{ik\theta_{\mu}}P_{l}^{m}(z_{\mu}\cdot
z)=0,\qquad(k,l)\in J_{f},\qquad z\in S^{m}.$

Let $M$ be a maximal subset of $\\{1,2,\ldots,n\\}$ that indexes the distinct
elements among the $z_{j}$. Writing $M_{j}:=\\{\mu\colon z_{\mu}=z_{j}\\}$,
$j\in M$, the previous equality becomes

$\displaystyle\sum_{j\in M}\left(\sum_{\mu\in
M_{j}}c_{\mu}e^{ik\theta_{\mu}}\right)P_{l}^{m}(z_{j}\cdot z)=0,\qquad(k,l)\in
J_{f},\qquad z\in S^{m}.$

In particular,

$\displaystyle\sum_{j\in M}\left(\sum_{\mu\in
M_{j}}c_{\mu}e^{ik\theta_{\mu}}\right)P_{l}^{m}(z_{j}\cdot z)=0,\qquad
l\in\\{l\colon(k,l)\in J_{f}\\},\qquad z\in S^{m},$

whenever $k\in\\{k\colon\\{l\colon(k,l)\in J_{f}\\}\in{\rm SPD}(S^{m})\\}$.
Since the $z_{j}$ in the expression above are all distinct, Lemma 2.2 yields
that

$\displaystyle\sum_{\mu\in M_{j}}c_{\mu}e^{ik\theta_{\mu}}=0,\qquad
k\in\big{\\{}k\colon\\{l\colon(k,l)\in J_{f}\\}\in{\rm
SPD}\big{(}S^{m}\big{)}\big{\\}},$

for every $j\in M$. An application of Lemma 2.3 for each $j$ plus the help of
our original assumption leads to $c_{\mu}=0$, $\mu\in M_{j}$, $j\in M$. But
this corresponds to $c=0$. ∎

In a similar manner, but with slightly easier arguments, the following cousin
theorem can be proved.

######  Theorem 2.5.

Let $f$ be the generating function of an isotropic and positive definite
kernel on $S^{1}\times S^{m}$ and consider its series representation (1.1). If

$\displaystyle\big{\\{}l\colon\\{k\colon(k,l)\in J_{f}\\}\in{\rm
SPD}\big{(}S^{1}\big{)}\big{\\}}\in{\rm SPD}\big{(}S^{m}\big{)},$

then the kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot
w)$ is strictly positive definite on $S^{1}\times S^{m}$.

We close this section presenting realizations for the previous theorems. They
follow from the characterizations for strict positive definiteness on singles
spheres described in the Introduction.

######  Corollary 2.6.

Let $f$ be the generating function for an isotropic and positive definite
kernel on $S^{1}\times S^{m}$ and consider its series representation (1.1).
Either condition below is sufficient for the kernel
$((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$ to be
strictly positive definite:

  1. $(i)$

$J_{f}$ contains sequences
$\\{(k_{\mu},2l_{\mu\nu})\colon\mu,\nu\in\mathbb{Z}_{+}\\}$ and
$\\{(k^{\prime}_{\mu},2l^{\prime}_{\mu\nu}+1)\colon\mu,\nu\in\mathbb{Z}_{+}\\}$
so that

$\displaystyle\\{\pm
k_{\mu}\colon\mu\in\mathbb{Z}_{+}\\}\cap(n\mathbb{Z}+j)\neq\varnothing,\qquad
j=0,1,\ldots,n-1,\qquad n\geq 1,$ $\displaystyle\\{\pm
k^{\prime}_{\mu}\colon\mu\in\mathbb{Z}_{+}\\}\cap(n\mathbb{Z}+j)\neq\varnothing,\qquad
j=0,1,\ldots,n-1,\qquad n\geq 1,$

and

$\displaystyle\lim_{\nu\to\infty}l_{\mu\nu}=\lim_{\nu\to\infty}l^{\prime}_{\mu\nu}=\infty,\qquad\mu\in\mathbb{Z}_{+}.$

  2. $(ii)$

$J_{f}$ contains sequences
$\\{(k_{\mu\nu},2l_{\mu})\colon\mu,\nu\in\mathbb{Z}_{+}\\}$ and
$\\{(k^{\prime}_{\mu\nu},2l^{\prime}_{\mu}+1)\colon\mu,\nu\in\mathbb{Z}_{+}\\}$
so that

$\displaystyle\\{\pm
k_{\mu\nu}\colon\nu\in\mathbb{Z}_{+}\\}\cap(n\mathbb{Z}+j)\neq\varnothing,\qquad
j=0,1,\ldots,n-1,\qquad n\geq 1,\qquad\mu\in\mathbb{Z}_{+},$
$\displaystyle\\{\pm
k^{\prime}_{\mu\nu}\colon\nu\in\mathbb{Z}_{+}\\}\cap(n\mathbb{Z}+j)\neq\varnothing,\qquad
j=0,1,\ldots,n-1,\qquad n\geq 1,\qquad\mu\in\mathbb{Z}_{+},$

and

$\displaystyle\lim_{\mu\to\infty}l_{\mu}=\lim_{\mu\to\infty}l^{\prime}_{\mu}=\infty.$

Finally, we would like to point that the previous theorems can be reproduced
in other settings, with or without the presence of isotropy (for example
$S^{m}\times S^{m}$ and the product of $S^{m}$ and a torus). Details will not
be included here.

##  3 Characterizations for strict positive definiteness on
$\boldsymbol{S^{1}\times S^{m}}$

In order to obtain a characterization for the strict positive definiteness of
an isotropic positive definite kernel on $S^{1}\times S^{m}$, we need to look
at the concept of strict positive definiteness in an enhanced form. We begin
this section explaining what we mean by that and introducing the additional
concepts needed.

It is an obvious matter to see that we can write the generating function $f$
of a positive definite kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to
f(x\cdot y,z\cdot w)$ in the form

$\displaystyle f(t,s)=\sum_{l=0}^{\infty}f_{l}(t)P_{l}^{m}(s),\qquad
t,s\in[-1,1],$ (3.1)

in which

$\displaystyle f_{l}(t):=\sum_{k=0}^{\infty}a_{k,l}^{1,m}P_{k}^{1}(t),\qquad
t\in[-1,1],\qquad l=0,1,\ldots.$

Since $P_{l}^{m}(1)\geq 1$, $m\geq 2$, $l=0,1,\ldots$, the series
$\sum\limits_{k=0}^{\infty}a_{k,l}^{1,m}P_{k}^{1}(1)<\infty$ converges. In
particular, each $f_{l}$ is the generating function of a continuous, isotropic
and positive definite kernel on $S^{1}$.

In the statement of the next lemma, we will employ the following additional
notation related to another one we have previously introduced:

$\displaystyle J_{f}^{k}:=\\{l\colon(k,l)\in J_{f}\\}.$

In particular,

$\displaystyle\cup_{k}J_{f}^{k}=\\{l\colon f_{l}\neq 0\\}.$

Among other things, the lemma suggests how a characterization for the strict
positive definiteness of an isotropic and positive definite kernel on
$S^{1}\times S^{m}$ should look like.

######  Lemma 3.1.

Let $f$ be the generating function of an isotropic and positive definite
kernel on $S^{1}\times S^{m}$ and consider the alternative series
representation (3.1) for $f$. If $p$ is a positive integer,
$x_{1},x_{2},\ldots,x_{p}$ are distinct points on $S^{1}$ and $c$ is a real
vector in $\mathbb{R}^{p}$, then the continuous function $g$ given by

$\displaystyle
g(s)=\sum_{l\in\cup_{k}J_{f}^{k}}\big{\\{}c^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}c\big{\\}}P_{l}^{m}(s),\qquad s\in[-1,1],$

generates an isotropic and positive definite kernel on $S^{m}$. If $c$ is
nonzero and the kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot
y,z\cdot w)$ is strictly positive definite, then $(z,w)\in(S^{m})^{2}\to
g(z\cdot w)$ is strictly positive definite as well.

###### Proof.

Write $c=(c_{1},c_{2},\ldots,c_{p})$. If $z_{1},z_{2},\ldots,z_{q}$ are
distinct points on $S^{m}$ and $d_{1},d_{2},\ldots,d_{q}$ are real numbers,
then

$\displaystyle\sum_{\mu,\nu=1}^{q}d_{\mu}d_{\nu}g(z_{\mu}\cdot
z_{\nu})=\sum_{\mu,\nu=1}^{q}\sum_{i,j=1}^{p}(d_{\mu}c_{i})(d_{\nu}c_{j})f(x_{i}\cdot
x_{j},z_{\mu}\cdot z_{\nu}).$

But, the last expression above corresponds to a quadratic form involving $f$
and the $pq$ distinct points $(x_{i},z_{\mu})$, $i=1,2,\ldots,p$,
$\mu=1,2,\ldots,q$, of $S^{1}\times S^{m}$. In particular,

$\displaystyle\sum_{\mu,\nu=1}^{q}d_{\mu}d_{\nu}g(z_{\mu}\cdot z_{\nu})\geq
0.$

If the real numbers $d_{\mu}$ are not all zero and $c\neq 0$, then at least
one of the scalars $d_{\mu}c_{i}$ is likewise nonzero. Further, if
$((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$ is strictly
positive definite, then the quadratic form above is, in fact, positive. In
particular, $(z,w)\in(S^{m})^{2}\to g(z\cdot w)$ is strictly positive
definite. ∎

Another useful technical result is as follows.

######  Lemma 3.2.

Let $f$ be the generating function of an isotropic and positive definite
kernel on $S^{1}\times S^{m}$ and consider the alternative series
representation (3.1) for $f$. If $(x_{1},z_{1}),(x_{2},z_{2}),$
$\ldots,(x_{n},z_{n})$ are distinct points in $S^{1}\times S^{m}$ and
$c_{1},c_{2},\ldots,c_{n}$ are real scalars, then the following assertions are
equivalent:

  1. $(i)$

$\sum\limits_{i,j=1}^{n}c_{i}c_{j}f(x_{i}\cdot x_{j},z_{i}\cdot z_{j})=0$;

  2. $(ii)$

$\sum\limits_{i,j=1}^{n}c_{i}c_{j}f_{l}(x_{i}\cdot x_{j})P_{l}^{m}(z_{i}\cdot
z_{j})=0$, $l\in\cup_{k}J_{f}^{k}$.

###### Proof.

One direction is immediate while the other follows simply from the observation
that $(t,s)\in[-1,1]^{2}\to f_{l}(t)P_{l}^{m}(s)$ is the generating function
of a positive definite kernel on $S^{1}\times S^{m}$. ∎

Next, we formalize the definition of enhancement we use in this section.

######  Definition 3.3.

Let $p$ and $q$ be positive integers, $\\{x_{1},x_{2},\ldots,x_{p}\\}\subset
S^{1}$ and $\\{z_{1},z_{2},\ldots,z_{q}\\}$ an antipodal free subset of
$S^{m}$, that is, a set containing no pairs of antipodal points. An enhanced
subset of $S^{1}\times S^{m}$ generated by them is the set

$\displaystyle\big{\\{}(x_{1},z_{1}),(x_{2},z_{1}),\ldots,(x_{p},z_{1}),(x_{1},z_{2}),(x_{2},z_{2}),\ldots,(x_{p},z_{2}),\ldots,$
$\displaystyle\qquad{}(x_{1},z_{q}),(x_{2},z_{q}),\ldots,(x_{p},z_{q}),(x_{1},-z_{1}),(x_{2},-z_{1}),\ldots,(x_{p},-z_{1}),$
$\displaystyle\qquad{}(x_{1},-z_{2}),(x_{2},-z_{2}),\ldots,(x_{p},-z_{2}),\ldots,(x_{1},-z_{q}),(x_{2},-z_{q}),\ldots,(x_{p},-z_{q})\big{\\}}.$

The positive numbers $p$ and $q$ in the previous definition may have no
connection at all. The order in which the elements in an enhanced subset of
$S^{1}\times S^{m}$ are displayed is not relevant, but the writing of the
upcoming results will take into account the order adopted above and inherited
from those in the subsets of $S^{1}$ and $S^{m}$ involved. An enhanced set as
above contains $2pq$ distinct points.

The following lemma is concerned with the existence of enhanced sets.

######  Lemma 3.4.

If
$B^{\prime}=\\{(x_{1}^{\prime},z_{1}^{\prime}),(x_{2}^{\prime},z_{2}^{\prime}),\ldots,(x_{n}^{\prime},z_{n}^{\prime})\\}$
is a subset of $S^{1}\times S^{m}$, then there exists an enhanced subset $B$
of $S^{1}\times S^{m}$ that contains $B^{\prime}$.

###### Proof.

It suffices to consider the enhanced subset of $S^{1}\times S^{m}$ generated
by the set $\\{x_{1},x_{2}$, $\ldots,x_{p}\\}$ that encompasses the distinct
elements among the $x_{i}^{\prime}$ and an antipodal free subset
$\\{z_{1},z_{2},\ldots,z_{q}\\}$ of $S^{m}$ satisfying
$z_{i}^{\prime}\in\\{z_{1},z_{2},\ldots,z_{q}\\}$, $i=1,2,\ldots,n$. ∎

If $f$ is the generating function of an isotropic and positive definite kernel
on $S^{1}\times S^{m}$ and $B$ is an enhanced set as previously described, we
will write $\mathcal{E}(f,B)$ to denote the interpolation matrix of
$((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$ at $B$,
keeping the order for the points of $B$. If $A$ is a subset of $S^{1}\times
S^{m}$ and $B$ is an enhanced subset of $S^{1}\times S^{m}$ containing $A$,
then the interpolation matrix of $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to
f(x\cdot y,z\cdot w)$ at $A$ is a principal sub-matrix of $\mathcal{E}(f,B)$.
In particular, if $\mathcal{E}(f,B)$ is positive definite, so is $A$. These
comments justify the following lemma.

######  Lemma 3.5.

Let $f$ be the generating function of an isotropic and positive definite
kernel on $S^{1}\times S^{m}$ and consider the alternative series
representation (3.1) for $f$. The following assertions are equivalent:

  1. $(i)$

the kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$
is strictly positive definite;

  2. $(ii)$

If $B$ is an enhanced subset of $S^{1}\times S^{m}$, then the matrix
$\mathcal{E}(f,B)$ is positive definite.

Due to the decomposition for the generating function $f$ of an isotropic
positive definite kernel as described in (3.1), we can write

$\displaystyle\mathcal{E}(f,B)=\sum_{l=0}^{\infty}\mathcal{E}(f,B,l)$

in which $\mathcal{E}(f,B,l)$ is the interpolation matrix of the kernel
$(t,s)\in S^{1}\times S^{m}\to f_{l}(t)P_{l}^{m}(s)$ at $B$. The order in
which the elements of an enhanced subset of $S^{1}\times S^{m}$ appears forces
the matrix $\mathcal{E}(f,B,l)$ to have a very distinctive block
representation. Precisely,

$\displaystyle\mathcal{E}(f,B,l)=\left(\begin{matrix}M_{11}&M_{12}\\\
M_{21}&M_{22}\end{matrix}\right),$

where each block $M_{\rho\sigma}=M_{\rho\sigma}(f,B,l)$ has its own block
structure

$\displaystyle
M_{\rho\sigma}=[M_{\rho\sigma}^{\mu\nu}]_{\mu,\nu=1}^{q},\qquad\rho,\sigma=1,2,$

defined by

$\displaystyle M_{\rho\sigma}^{\mu\nu}=[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}(-1)^{l(\sigma+\rho)}P_{l}^{m}(z_{\mu}\cdot
z_{\nu}),\qquad\mu,\nu=1,2,\ldots,q.$

Implicitly used in the writing of the block decomposition above is the fact
that Gegenbauer polynomials of even degree are even functions while those of
odd degree are odd functions. In particular, since $M_{22}=M_{11}$ and
$M_{12}=M_{21}=(-1)^{l}M_{11}$, the matrix $\mathcal{E}(f,B,l)$ depends upon
$M_{11}$ only.

Keeping all the notation introduced so far, Lemmas 3.2 and 3.5 and the
comments above lead to the following characterization for the strict positive
definiteness of the kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to
f(x\cdot y,z\cdot w)$ via

$\displaystyle M_{11}=\big{[}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}P_{l}^{m}(z_{\mu}\cdot z_{\nu})\big{]}_{\mu,\nu=1}^{q}.$

######  Lemma 3.6.

Let $f$ be the generating function of an isotropic and positive definite
kernel on $S^{1}\times S^{m}$ and consider the alternative series
representation (3.1) for $f$. The following assertions are equivalent:

  1. $(i)$

the kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$
is strictly positive definite;

  2. $(ii)$

if $B$ is an enhanced subset of $S^{1}\times S^{m}$ generated by a subset
$\\{x_{1},x_{2},\ldots,x_{p}\\}$ of $S^{1}$ and the antipodal free subset
$\\{z_{1},z_{2},\ldots,z_{q}\\}$ of $S^{m}$, then the only solution
$(c_{1},c_{2})\in(\mathbb{R}^{pq})^{2}$ of the system

$\displaystyle\big{[}c_{1}+(-1)^{l}c_{2}\big{]}^{t}M_{11}\big{[}c_{1}+(-1)^{l}c_{2}\big{]}=0,\qquad
l\in\cup_{k}J_{f}^{k},$

is the trivial one, that is, $c_{1}=c_{2}=0$.

Introducing components for the vectors $c_{i}$ in the previous lemma, we
obtain the following reformulation.

######  Lemma 3.7.

Let $f$ be as in the previous lemma. The following assertions are equivalent:

  1. $(i)$

the kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$
is strictly positive definite;

  2. $(ii)$

if $B$ is an enhanced subset of $S^{1}\times S^{m}$ generated by a subset
$\\{x_{1},x_{2},\ldots,x_{p}\\}$ of $S^{1}$ and the antipodal free subset
$\\{z_{1},z_{2},\ldots,z_{q}\\}$ of $S^{m}$, then the only solution
$(c_{1}^{1},c_{1}^{2},\ldots,c_{1}^{q},c_{2}^{1},c_{2}^{2},$
$\ldots,c_{2}^{q})\in(\mathbb{R}^{p})^{2q}$ of the system

$\displaystyle\sum_{\mu,\nu=1}^{q}\big{\\{}(c_{1}^{\mu}+(-1)^{l}c_{2}^{\mu})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}(c_{1}^{\nu}+(-1)^{l}c_{2}^{\nu})\big{\\}}P_{l}^{m}(z_{\mu}\cdot
z_{\nu})=0,\qquad l\in\cup_{k}J_{f}^{k},$

is the trivial one.

Next, we break up the system in the previous lemma, according to the parity of
the elements in $\cup_{k}J_{f}^{k}$.

######  Proposition 3.8.

Let $f$ be as in the previous lemma. The following assertions are equivalent:

  1. $(i)$

the kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$
is strictly positive definite;

  2. $(ii)$

if $p$ and $q$ are positive integers, $x_{1},x_{2},\ldots,x_{p}$ are distinct
points on $S^{1}$ and $\\{z_{1},z_{2},\ldots,$ $z_{q}\\}$ is an antipodal free
subset of $S^{m}$, then the only solution
$(d_{1}^{1},d_{1}^{2},\ldots,d_{1}^{q},d_{2}^{1},d_{2}^{2},\ldots,d_{2}^{q})$
in $(\mathbb{R}^{p})^{2q}$ of the system

$\displaystyle\sum_{\mu,\nu=1}^{q}\big{\\{}(d_{1}^{\mu})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{1}^{\nu}\big{\\}}P_{l}^{m}(z_{\mu}\cdot
z_{\nu})=0,\qquad l\in(2\mathbb{Z}_{+}+1)\cap\big{(}\cup_{k}J_{f}^{k}\big{)},$
$\displaystyle\sum_{\mu,\nu=1}^{q}\big{\\{}(d_{2}^{\mu})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\nu}\big{\\}}P_{l}^{m}(z_{\mu}\cdot
z_{\nu})=0,\qquad l\in(2\mathbb{Z}_{+})\cap\big{(}\cup_{k}J_{f}^{k}\big{)},$

is the trivial one.

###### Proof.

Assume that for some distinct points $x_{1},x_{2},\ldots,x_{p}$ in $S^{1}$ and
some antipodal free subset $\\{z_{1},z_{2},\ldots,z_{q}\\}$ of $S^{m}$, the
system in $(ii)$ has a nontrivial solution
$(d_{1}^{1},d_{1}^{2},\ldots,d_{1}^{q},d_{2}^{1},$
$d_{2}^{2},\ldots,d_{2}^{q})$. If $d_{1}^{\mu}\neq 0$ for some $\mu$, we
define $c_{1}^{\mu}=-c_{2}^{\mu}=2^{-1}d_{1}^{\mu}$, $\mu=1,2,\ldots,q$.
Otherwise, we define $c_{1}^{\mu}=c_{2}^{\mu}=2^{-1}d_{2}^{\mu}$,
$\mu=1,2,\ldots,q$. In both cases, the vector
$(c_{1}^{1},c_{1}^{2},\ldots,c_{1}^{q},c_{2}^{1},c_{2}^{2},\ldots,c_{2}^{q})$
is nonzero and, in addition,

$\displaystyle\sum_{\mu,\nu=1}^{q}\big{\\{}\big{(}c_{1}^{\mu}+(-1)^{l}c_{2}^{\mu}\big{)}^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}\big{(}c_{1}^{\nu}+(-1)^{l}c_{2}^{\nu}\big{)}\big{\\}}P_{l}^{m}(z_{\mu}\cdot
z_{\nu})=0,$

for
$l\in[(2\mathbb{Z}_{+}+1)\cap(\cup_{k}J_{f}^{k})]\cup[(2\mathbb{Z}_{+})\cap(\cup_{k}J_{f}^{k})]=\cup_{k}J_{F}^{k}$.
In other words, Condition $(ii)$ in Lemma 3.7 does not hold for the enhanced
set $B$ generated by the subset $\\{x_{1},x_{2},\ldots,x_{p}\\}$ of $S^{1}$
and the antipodal free subset $\\{z_{1},z_{2},\ldots,z_{q}\\}$ of $S^{m}$.
Thus, $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$ is not
strictly positive definite. Conversely, if $(i)$ does not hold, the previous
lemma assures the existence of a subset $\\{x_{1},x_{2},\ldots,x_{p}\\}$ of
$S^{1}$, an antipodal free subset $\\{z_{1},z_{2},\ldots,z_{q}\\}$ of $S^{m}$,
an enhanced subset $A$ of $S^{1}\times S^{m}$ generated by them and a nonzero
vector
$(c_{1}^{1},c_{1}^{2},\ldots,c_{1}^{q},c_{2}^{1},c_{2}^{2},\ldots,c_{2}^{q})\in(\mathbb{R}^{p})^{2q}$
so that

$\displaystyle\sum_{\mu,\nu=1}^{q}\big{\\{}\big{(}c_{1}^{\mu}+(-1)^{l}c_{2}^{\mu}\big{)}^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}\big{(}c_{1}^{\nu}+(-1)^{l}c_{2}^{\nu}\big{)}\big{\\}}P_{l}^{m}(z_{\mu}\cdot
z_{\nu})=0,\qquad l\in\cup_{k}J_{f}^{k}.$

However, this last piece of information corresponds to

$\displaystyle\sum_{\mu,\nu=1}^{q}\big{\\{}\big{(}c_{1}^{\mu}-c_{2}^{\mu}\big{)}^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}\big{(}c_{1}^{\nu}-c_{2}^{\nu}\big{)}\big{\\}}P_{l}^{m}(z_{\mu}\cdot
z_{\nu})=0,\qquad l\in(2\mathbb{Z}_{+}+1)\cap\big{(}\cup_{k}J_{f}^{k}\big{)},$

and

$\displaystyle\sum_{\mu,\nu=1}^{q}\big{\\{}\big{(}c_{1}^{\mu}+c_{2}^{\mu}\big{)}^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}\big{(}c_{1}^{\nu}+c_{2}^{\nu}\big{)}\big{\\}}P_{l}^{m}(z_{\mu}\cdot
z_{\nu})=0,\qquad l\in(2\mathbb{Z}_{+})\cap\big{(}\cup_{k}J_{f}^{k}\big{)}.$

On the other hand, it is easily verifiable that the vector

$\displaystyle\big{(}c_{1}^{1}-c_{2}^{1},c_{1}^{2}-c_{2}^{2},\ldots,c_{1}^{q}-c_{2}^{q},c_{1}^{1}+c_{2}^{1},c_{1}^{2}+c_{2}^{2},\ldots,c_{1}^{q}+c_{2}^{q}\big{)}\in\big{(}\mathbb{R}^{p}\big{)}^{2q}$

is nonzero. Thus, $(ii)$ does not hold due to Lemma 3.7. ∎

We are about ready to prove the crucial result in this section.

######  Theorem 3.9.

Let $f$ be the generating function of an isotropic and positive definite
kernel on $S^{1}\times S^{m}$ and consider the alternative series
representation (3.1) for $f$. The following assertions are equivalent:

  1. $(i)$

the kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$
is strictly positive definite;

  2. $(ii)$

if $p$ is a positive integer, $x_{1},x_{2},\ldots,x_{p}$ are distinct points
in $S^{1}$ and $c\in\mathbb{R}^{p}\setminus\\{0\\}$, then the set

$\displaystyle\big{\\{}l\in\cup_{k}J_{f}^{k}\colon c^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}c>0\big{\\}}$

contains infinitely many even and infinitely many odd integers.

###### Proof.

Lemma 3.1 justifies one implication. As for the other, assume the condition in
the statement of the theorem holds but $((x,z),(y,w))\in(S^{1}\times
S^{m})^{2}\to f(x\cdot y,z\cdot w)$ is not strictly positive definite. Hence,
we can find a positive integer $p$, distinct points $x_{1},x_{2},\ldots,x_{p}$
in $S^{1}$, an antipodal free subset $\\{y_{1},y_{2},\ldots,y_{n}\\}$ of
$S^{m}$ and a nonzero vector
$(d_{1}^{1},d_{1}^{2},\ldots,d_{1}^{q},d_{2}^{1},d_{2}^{2},\ldots,d_{2}^{q})$
in $(\mathbb{R}^{p})^{2q}$ so that the two equations in Proposition 3.8(ii)
hold. We will proceed assuming that $(d_{2}^{1},d_{2}^{2},\ldots,d_{2}^{q})$
is a nonzero vector and that

$\displaystyle\sum_{\mu,\nu=1}^{q}\big{\\{}(d_{2}^{\mu})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\nu}\big{\\}}P_{l}^{m}(z_{\mu}\cdot
z_{\nu})=0,\qquad l\in 2\mathbb{Z}_{+}\cap\big{(}\cup_{k}J_{f}^{k}\big{)}$

and will reach a contradiction. The other possibility can be handled similarly
but the details will be omitted. Without loss of generality, we can assume
that the vector $d_{2}^{1}$ is nonzero. Since the set

$\displaystyle\big{\\{}l\in\cup_{k}J_{f}^{k}\colon\big{(}d_{2}^{1}\big{)}^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{1}>0\big{\\}}\cap 2\mathbb{Z}$

is infinite by assumption, we can select an infinite subset $Q$ of it and a
number $\theta=\theta(l)$ in $\\{1,2,\ldots,q\\}$ so that

$\displaystyle\big{(}d_{2}^{\theta}\big{)}^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\theta}\geq\big{(}d_{2}^{\mu}\big{)}^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\mu},\qquad\mu=1,2,\ldots,q,\qquad l\in Q.$

In particular,

$\displaystyle\big{(}d_{2}^{\theta}\big{)}^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\theta}>0,\qquad l\in Q.$

Returning to the initial equality we can write

$\displaystyle 0=1+\sum_{\begin{subarray}{c}\mu=1\\\
\mu\neq\theta\end{subarray}}^{q}\frac{(d_{2}^{\mu})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\mu}}{(d_{2}^{\theta})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\theta}}\frac{P_{l}^{m}(z_{\mu}\cdot
z_{\mu})}{P_{l}^{m}(1)}$
$\displaystyle\hphantom{0=1}{}+\sum_{\mu\neq\nu}\frac{(d_{2}^{\mu})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\nu}}{(d_{2}^{\theta})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\theta}}\frac{P_{l}^{m}(z_{\mu}\cdot
z_{\nu})}{P_{l}^{m}(1)},\qquad l\in Q.$

Since each $f_{l}$ is the continuous and isotropic part of a positive definite
kernel on $S^{1}$, we have that $(d_{2}^{\mu})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\mu}\geq 0$, $\mu=1,2,\ldots,q$. In particular,

$\displaystyle\sum_{\begin{subarray}{c}\mu=1\\\
\mu\neq\theta\end{subarray}}^{q}\frac{(d_{2}^{\mu})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\mu}}{(d_{2}^{\theta})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\theta}}\in[0,q-1],$

while the Cauchy–Schwarz inequality implies that

$\displaystyle 0\leq\left|\frac{(d_{2}^{\mu})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\nu}}{(d_{2}^{\theta})^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}d_{2}^{\theta}}\right|\leq 1,\qquad\mu\neq\nu.$

Since $z_{\mu}\cdot z_{\nu}\in(-1,1)$, $\mu\neq\nu$, a well-known property of
the Gegenbauer polynomials provides the limit formula [20, p. 196]

$\displaystyle\lim_{\begin{subarray}{c}l\to\infty\\\ l\in
Q\end{subarray}}\frac{P_{l}^{m}(z_{\mu}\cdot
z_{\nu})}{P_{l}^{m}(1)}=0,\qquad\mu\neq\nu.$

Consequently, we may apply the definition of limit conveniently ($l$ large
enough), to conclude that $0\geq 1+0-1/2=1/2$, a contradiction. ∎

The next theorem demands the truncated sum functions ($\gamma\geq 0$)

$\displaystyle f_{\gamma}^{\rm
o}=\sum_{2l+1\geq\gamma}f_{2l+1}\qquad\mbox{and}\qquad f_{\gamma}^{\rm
e}=\sum_{2l\geq\gamma}f_{2l}$

attached to the generating function of an isotropic and positive definite
kernel. Since $P_{l}^{m}(1)\geq 1$, $m\geq 2$, $l\geq 0$, it follows that

$\displaystyle|f_{l}(t)|\leq\sum_{k=0}^{\infty}a_{k,l}^{1,m}P_{k}^{1}(1)\leq\sum_{k=0}^{\infty}a_{k,l}^{1,m}P_{k}^{1}(1)P_{l}^{m}(1),\qquad
l\geq\gamma.$

In particular, since
$\sum\limits_{l=0}^{\infty}\sum\limits_{k=0}^{\infty}a_{k,l}^{1,m}P_{k}^{1}(1)P_{l}^{m}(1)<\infty$,
the functions $f_{\gamma}^{\rm o}$ and $f_{\gamma}^{\rm e}$ are continuous.

######  Theorem 3.10.

Let $f$ be the generating function of an isotropic and positive definite
kernel on $S^{1}\times S^{m}$ and consider the alternative series
representation (3.1) for $f$. The following assertions are equivalent:

  1. $(i)$

the kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$
is strictly positive definite;

  2. $(ii)$

for each $\gamma\geq 0$, the functions $f_{\gamma}^{\rm o}$ and
$f_{\gamma}^{\rm e}$ are the generating functions of isotropic and strictly
positive definite kernels on $S^{1}$.

###### Proof.

If $(i)$ holds, we can apply Lemma 3.1 in order to see that

$\displaystyle c^{t}f_{\gamma}^{\rm o}(x_{i}\cdot
x_{j})c=\sum_{2l+1\geq\gamma}c^{t}f_{2l+1}(x_{i}\cdot x_{j})c>0,$

whenever $c\in\mathbb{R}^{p}\setminus\\{0\\}$, $\gamma\geq 0$ and
$x_{1},x_{2},\ldots,x_{p}$ are distinct points in $S^{1}$. Obviously, a
similar property is valid for $f_{\gamma}^{\rm e}$. Thus, $(ii)$ follows.
Conversely, if $(ii)$ holds, then Condition $(ii)$ in the previous theorem
holds as well, due to the fact that $(ii)$ is valid for all $\gamma\geq 0$.
Thus, Theorem 3.9 guarantees the strict positive definiteness of
$((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$. ∎

The coefficient in the series expansion of $f_{\gamma}^{\rm o}$ attached to
the polynomial $P_{k}^{1}$ is

$\displaystyle\sum_{\gamma\leq 2l+1\in J_{f}^{k}}a_{k,2l+1}^{1,m}.$

It is positive if, and only if, $J_{f}^{k}$ contains an odd integer greater
than or equal to $\gamma$. A similar remark applies to the coefficients in the
series of $f_{\gamma}^{\rm e}$. Taking into account the characterization for
strict positive definiteness on $S^{1}$ quoted at the introduction, we have
the following consequence of the previous theorem and our final
characterization for strict positive definiteness on $S^{1}\times S^{m}$.

######  Theorem 3.11.

Let $f$ be the generating function of an isotropic and positive definite
kernel on $S^{1}\times S^{m}$. The following assertions are equivalent:

  1. $(i)$

the kernel $((x,z),(y,w))\in(S^{1}\times S^{m})^{2}\to f(x\cdot y,z\cdot w)$
is strictly positive definite;

  2. $(ii)$

for each $\gamma\geq 0$, the sets

$\displaystyle\big{\\{}k\in\mathbb{Z}\colon
J_{f}^{|k|}\cap\\{\gamma,\gamma+1,\ldots\\}\cap(2\mathbb{Z}+1)\neq\varnothing\big{\\}}$

and

$\displaystyle\big{\\{}k\in\mathbb{Z}\colon
J_{f}^{|k|}\cap\\{\gamma,\gamma+1,\ldots\\}\cap
2\mathbb{Z}_{+}\neq\varnothing\big{\\}}$

intersect every arithmetic progression in $\mathbb{Z}$.

##  4 Replacing $\boldsymbol{S^{m}}$ with a compact two-point homogeneous
space

The results obtained so far in the paper can be adapted to hold for kernels on
a product of the form $S^{1}\times\mathbb{M}^{d}$, in which $\mathbb{M}^{d}$
is a compact two-point homogeneous space. The case $S^{1}\times S^{1}$ was
covered in [13] while the case $S^{d}\times\mathbb{M}^{d}$, $d\geq 3$, was
covered in [2, Theorem 4.5]. The results sketched in this section complement
these two cases.

Let us write $|zw|$ to denote the usual normalized surface distance between
$z$ and $w$ in $\mathbb{M}^{d}$. As described in [2], an isotropic kernel
$((x,z),(y,w))\in(S^{1}\times\mathbb{M}^{d})^{2}\to f(x\cdot y,\cos(|zw|/2))$
is positive definite if, and only if, the generating function $f$ has a double
series representation in the form

$\displaystyle
f(t,s)=\sum_{k,l=0}^{\infty}a_{k,l}^{d}P_{k}^{1}(t)P_{l}^{d,\beta}(s),\qquad
t,s\in[-1,1]^{2},$

in which $a_{k,l}^{d}\geq 0$, $k,l\in\mathbb{Z}_{+}$, $P_{l}^{d,\beta}$ is the
Jacobi polynomial associated to the pair $((d-2)/2,\beta)$, $\beta$ is a
number from the list $-1/2,0,1,3$, depending on the respective category
$\mathbb{M}^{d}$ belongs to, that is, the real projective spaces
$\mathbb{P}^{d}(\mathbb{R})$, $d=2,3,\ldots$, the complex projective spaces
$\mathbb{P}^{d}(\mathbb{C})$, $d=4,6,\ldots$, the quaternionic projective
spaces $\mathbb{P}^{d}(\mathbb{H})$, $d=8,12,\ldots$, and the Cayley
projective plane $\mathbb{P}^{d}({\rm Cay})$, $d=16$, and
$\sum\limits_{k,l=0}^{\infty}a_{k,l}P_{k}^{1}(1)P_{l}^{d,\beta}(1)<\infty$.

In this setting, the procedure adopted in Section 3 can be considerably
simplified. An alternative series representation for the generating function
of the kernel can be likewise defined and the fact that a point in
$\mathbb{M}^{d}$ possesses infinitely many antipodal points permits the
deduction of a version of Theorem 3.9 without considering any enhancements and
augmentations. Precisely, we have the following result.

######  Theorem 4.1.

Let $f$ be the generating function of an isotropic and positive definite
kernel on $S^{1}\times\mathbb{M}^{d}$ and consider the alternative series
representation for $f$. The following assertions are equivalent:

  1. $(i)$

the kernel $((x,z),(y,w))\in(S^{1}\times\mathbb{M}^{d})^{2}\to f(x\cdot
y,z\cdot w)$ is strictly positive definite;

  2. $(ii)$

if $p$ is a positive integer, $x_{1},x_{2},\ldots,x_{p}$ are distinct points
in $S^{1}$ and $c\in\mathbb{R}^{p}\setminus\\{0\\}$, then the set

$\displaystyle\big{\\{}l\in\cup_{k}J_{f}^{k}\colon c^{t}[f_{l}(x_{i}\cdot
x_{j})]_{i,j=1}^{p}c>0\big{\\}}$

contains infinitely many integers.

Taking into account the characterization for strict positive definiteness
obtained in [1], the final characterization in these remaining cases is this
one.

######  Theorem 4.2.

Let $f$ be the generating function of an isotropic and positive definite
kernel on $S^{1}\times\mathbb{M}^{d}$. Assume $\mathbb{M}^{d}$ is not a
sphere. The following assertions are equivalent:

  1. $(i)$

the kernel $((x,z),(y,w))\in(S^{1}\times\mathbb{M}^{d})^{2}\to f(x\cdot
y,z\cdot w)$ is strictly positive definite;

  2. $(ii)$

for each $\gamma\geq 0$, the set

$\displaystyle\big{\\{}k\in\mathbb{Z}\colon\
J_{f}^{|k|}\cap\\{\gamma,\gamma+1,\ldots\\}\neq\varnothing\big{\\}}$

intersects every arithmetic progression in $\mathbb{Z}$.

### Acknowledgements

The authors are grateful to the referees for their valuable comments and
suggestions. Second author acknowledges partial financial support from FAPESP,
grant 2014/00277-5. Likewise, the third author acknowledges support from the
same foundation, under grants 2014/25796-5 and 2016/03015-7.

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