# The Foldy-Lax approximation is valid for nearly resonating frequencies

Abdulaziz Alsenafi∗, Ahcene Ghandriche ∗∗, Mourad Sini‡

###### Abstract.

The waves (including acoustic, electromagnetic and elastic ones) propagating
in the presence of a cluster of inhomogeneities undergo multiple interactions
between them. When these inhomogeneities have sub-wavelength sizes, the
dominating field due to the these multiple interactions is the Foldy-Lax
field. This field models the interaction between the equivalent point-like
scatterers, located at the centers of the small inhomogeneities, with
scattering coefficients related to geometrical/material properties of each
inhomogeneities, as polarization coefficients. One of the related questions
left open for a long time is whether we can reconstruct this Foldy-Lax field
from the scattered field measured far away from the cluster of the small
inhomogeneities. This is the Foldy-Lax approximation (or Foldy-Lax paradigm).
In this work, we show that this approximation is indeed valid as soon as the
inhomogeneities enjoy critical scales between their sizes and contrasts. These
critical scales allow them to generate resonances which can be characterized
and computed. The main result here is that exciting the cluster by incident
frequencies which are close to the real parts of these resonances allow us to
reconstruct the Fold-Lax field from the scattered waves collected far away
from the cluster itself (as the farfields). In short, we show that the Foldy-
Lax approximation is valid using nearly resonating incident frequencies. This
results is demonstrated by using, as small inhomogeneities, dielectric
nanoparticles for the $2$D TM model of electromagnetic waves and bubbles for
the $3$D acoustic waves.

###### Key words and phrases:

Foldy-Lax approximation, Multiple scattering, Resonances.

###### 2010 Mathematics Subject Classification:

35R30, 35C20

∗ Department of Mathematics, Faculty of Science, Kuwait University, P.O. Box
5969, Safat 13060, Kuwait. Email<EMAIL_ADDRESS>
∗∗ RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz,
Austria. Email<EMAIL_ADDRESS>
‡ RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz,
Austria. Email<EMAIL_ADDRESS>
##  1\. Introduction

We are interested in the following model

(1.1) $\begin{cases}\nabla\cdot\left(a\nabla u\right)+k^{2}b\;u&=0,\mbox{ in
}\mathbb{R}^{m},\,\,m=2\,\,\text{or}\,\,3,\\\
u(x,\theta)&:=u^{i}(x,\theta)+u^{s}(x,\theta)\\\ \displaystyle\frac{\partial
u^{s}}{\partial|x|}-i\;k\sqrt{\frac{b}{a}}\;u^{s}&=o\left(\displaystyle\frac{1}{|x|^{\frac{m-1}{2}}}\right),|x|\rightarrow\infty.\end{cases}$

where

(1.2) $a:=\begin{cases}a_{j}\mbox{ in }D_{j},\\\ a_{0}\mbox{ in
}\mathbb{R}^{m}\setminus\underset{j=1}{\overset{M}{\cup}}D_{j},\end{cases}\quad\text{and}\quad
b:=\begin{cases}b_{j}\mbox{ in }D_{j},\\\ b_{0}\mbox{ in
}\mathbb{R}^{m}\setminus\underset{j=1}{\overset{M}{\cup}}D_{j},\end{cases}$

with constant values $a_{j}$ and $b_{j}$, $j=0,1,...,M$. Here $u^{i}$ is an
incident wave, solution to $\Delta u^{i}+k^{2}\frac{b_{0}}{a_{0}}u^{i}=0$ in
the whole space $\mathbb{R}^{m}$. Mostly, we deal with plane waves of the form
$u^{i}:=u^{i}(x,\theta):=e^{i\;k\sqrt{\frac{b_{0}}{a_{0}}}\theta\cdot x}$ with
an incident direction $\theta,\;|\theta|=1$. In dimension $m=2$, this model is
related to the TM model in electromagnetism where $u(\cdot)$ is the third
component of the electric field while $a_{j}$’s are the inverses of the
magnetic permeability constants and $b_{j}$’s are the electric permittivity
constants. We can also see $u(\cdot)$ as the third component of the magnetic
field where, now, $b_{j}$’s are the inverses of the magnetic permeability
constants and $a_{j}$’s are the electric permittivity constants. The
inhomogeneities $D_{j}$’s model here nano-particles. In dimension $m=3$, this
model is related to the propagation of acoustic wave $u(\cdot)$ in the
presence of micro bubbles $D_{j}$’s. Here the constants $a_{j}$’s are the
inverses of the mass density and the constants $b_{j}$’s are the inverses of
the bulk moduli of the bubbles, [11, 12].

The inhomogeneities $D_{j}$’s are small scaled (i.e. micro scaled for the
bubbles and nano scaled for the nano-particles). We take them of the form
$D_{j}:=z_{j}+\delta\;B_{j}$, $j=1,...,M$, where the $B_{j}$’s are Lipschitz
smooth and bounded domains containing the origin (of $\mathbb{R}^{m}$),
$z_{j}$’s model their location and $\delta$ their relative radius which we
take to be small, i.e.
$\delta=o\left(\underset{j=1,\cdots,M}{\max}\\{\|B_{j}\|\\}\right)$. We set
$D:=\overset{M}{\underset{j=1}{\cup}}D_{j}$. Let
$u(\cdot)=u^{s}(\cdot)+u^{i}(\cdot)$ be the solution of the acoustic
scattering problem (1.1). From the Lippmann-Schwinger representation of the
total acoustic field $u(\cdot)$, we have

(1.3) $u(x)-\alpha\;\underset{x}{\mathrm{div}\,}\int_{D}\Phi_{k}(x,y)\nabla
u(y)dy-\beta k^{2}\int_{D}\Phi_{k}(x,y)u(y)dy=u^{i}(x),$

where $\alpha:=a_{1}-a_{0}$ and $\beta:=b_{1}-b_{0}$ represent the contrasts
between the inner and the outer acoustic coefficients. Here,
$\Phi_{k}(\cdot,\cdot)$ is the Green’s function of the background medium
$(a_{0},b_{0})$ satisfying the outgoing Sommerfeld radiation conditions at
infinity. This is an integro-differential equation. To transform it to a
solely integral equation, we proceed by integration by parts, then (1.3)
becomes:

$u(x)-\gamma k^{2}\int_{D}\Phi_{k}(x,y)u(y)dy+\alpha\int_{\partial
D}\Phi_{k}(x,y)\frac{\partial u}{\partial\nu}(y)dy=u^{i}(x),$

where $\gamma:=\beta-\alpha b_{1}/a_{1}$, for $x\in D$. In addition, taking
the normal derivative and trace, with the usual traces of the double layer
potential, we obtain:

$\left(1+\frac{\alpha}{2}\right)\frac{\partial u}{\partial\nu}-\gamma
k^{2}\partial_{\nu-}\int_{D}\Phi_{k}(x,y)u(y)dy+\alpha(K_{D}^{k})^{*}\left[\frac{\partial
u}{\partial\nu}\right]=\frac{\partial u^{i}}{\partial\nu}.$

Hence for $x\in\mathbb{R}^{3}\setminus\overline{D}$, the total acoustic field
$u(x)$ is characterized by $u_{|_{D}}$ and $\frac{\partial
u}{\partial\nu}\big{|}_{\partial D}$, which are solutions of the following
close form system of integral equations:

(1.4) $\displaystyle\left[I-\gamma k^{2}A^{k}\right]u+\alpha\int_{\partial
D}\Phi_{k}(x,y)\frac{\partial u}{\partial\nu}(y)dy$ $\displaystyle=$
$\displaystyle u^{i}(x),\;\mbox{in}\,\,D,$ (1.5)
$\displaystyle\left[\frac{1}{\alpha}+\frac{1}{2}+(K_{D}^{k})^{*}\right]\left[\frac{\partial
u}{\partial\nu}\right]-\frac{\gamma}{\alpha}k^{2}\partial_{\nu-}\int_{D}\Phi_{k}(x,y)u(y)dy$
$\displaystyle=$ $\displaystyle\frac{1}{\alpha}\frac{\partial
u^{i}}{\partial\nu},\;\mbox{on}\,\,\partial D,$

with the Newtonian (a volume-type) operator:

$A^{k}:\mathbb{L}^{2}(D)\longrightarrow\mathbb{L}^{2}(D),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
A^{k}(u)(x):=\int_{D}\Phi_{k}(x,y)u(y)dy,$

with image of $A^{k}$ in $\mathbb{H}^{2}(D)$, and the Neumann-Poincaré (a
surface-type) operator111The notation $p.v$ means the Cauchy principal value.

$(K_{D}^{k})^{*}:\mathbb{H}^{-1/2}(\partial
D)\longrightarrow\mathbb{H}^{-1/2}(\partial D),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
(K_{D}^{k})^{*}(f)(x):=p.v.\int_{\partial
D}\frac{\partial}{\partial\nu(x)}\Phi_{k}(x,y)f(y)d\sigma(y).$

We recall that for $k=0$, each of these operators generates a sequence of
eigenvalues: $\lambda_{m}(A^{0})\stackrel{{\scriptstyle
m\rightarrow\infty}}{{\longrightarrow}}0$ and
$\sigma_{p}((K_{D}^{0})^{*})\subset[-\frac{1}{2},\frac{1}{2})$. In addition,
we have $K^{0}_{D}(1)=-\,\frac{1}{2}$. These singular values are behind the
resonances that we want to use.

We consider the coefficients $a$ and $b$ satisfying the following scales in
terms of $\delta$. We provide the following situations where the constants
$a_{0}$ and $b_{0}$ in the background are assumed to be independent (or
uniform in terms of) of $\delta$.

  1.   2. (1)

First situation.

(1.6) $a:=\begin{cases}a_{j}\sim\delta^{-2}\quad\text{inside }D_{j},\\\
a_{0}\quad\text{outside
}\overset{M}{\underset{j=1}{\cup}}D_{j},\end{cases}\text{and}\qquad
b:=\begin{cases}b_{j}\sim\delta^{-2}\quad\text{inside }D_{j},\\\
b_{0}\quad\text{outside }\overset{M}{\underset{j=1}{\cup}}D_{j}.\end{cases}$

In this case, we have $\gamma\sim 1$ and then $\gamma\,k^{2}\,A^{k}\ll 1$ as
$a\ll 1$. Hence, there is no singularity coming from (1.4). But as $\alpha\gg
1$, precisely if $\alpha\sim a^{-2}$ as $a\ll 1$, then we can excite the
eigenvalue $-\frac{1}{2}$ of $K_{D}^{0}$ and create a singularity in (1.5). In
this case, we have the _Minnaert resonance_ with surface-modes. Its dominant
part has the following value

$k^{2}_{M}:=\sqrt{\frac{8\,\pi\,a_{0}}{a_{1}\,\Theta_{\partial
D}}}\,\,\,\text{with}\,\,\,\Theta_{\partial D}:=\frac{1}{\left|\partial
D\right|}\int_{\partial D}\int_{\partial
D}\frac{(x-y)\cdot\nu(x)}{4\pi|x-y|}\,d\sigma(x)d\sigma(y)=\delta^{2}\;\Theta_{\partial
B}.$

This resonance was first observed in [6] based on indirect integral equation
methods. This observation was used for different purposes, see [7, 8, 9]. This
result was extended to more general families of micro-bubbles in [4, 3].

  3. (2)

Second situation.
Observe that if $\alpha$ is negative as in the case of the Drude model for
electromagnetism (recall here that $a$ plays the role of the inverse of the
permeability and $b$ the one of the permittivity) i.e.

(1.7)
$\epsilon:=\epsilon_{0}-\frac{k_{p}^{2}}{k(k+i\;\gamma_{dp})}\leavevmode\nobreak\
\leavevmode\nobreak\
\left(\text{or}\;\mu:=\mu_{0}-\frac{k_{p}^{2}}{k(k+i\;\gamma_{dp})}\right),$

with $k_{p}$ is the plasma frequency, $\gamma_{dp}$ the damping parameter and
$k$ is our incident frequency. Then we could excite the other sequence of
eigenvalues, $\sigma_{n}$, of $(K_{D}^{0})^{*}$ by choosing $\gamma_{dp}$
small in terms of $\delta$ and $k$ near the plasma frequency. Indeed, the
operator $\displaystyle\frac{1}{\alpha}+\frac{1}{2}+(K_{D}^{0})^{*}$ is non
injective if $k$ solve the dispersion equations
$\displaystyle\frac{1}{\alpha}+\frac{1}{2}+\sigma_{p}((K_{D}^{0})^{*})=0$
which we write in terms of the Drude model as
$\displaystyle\frac{\epsilon_{0}\,\epsilon}{\epsilon_{0}-\epsilon}+\frac{1}{2}+\sigma_{p}((K_{D}^{0})^{*})=0$
which gives, using $(\ref{Second})$ the dispersion equations
$\displaystyle\frac{\epsilon_{0}\left[k(k+i\,\gamma_{dp})\epsilon_{0}-k_{p}^{2}\right]}{k^{2}_{p}}+\frac{1}{2}+\sigma_{n}=0$.
As $\frac{1}{2}+\sigma_{n}\geq 0$, then we should take the frequency
$k^{2}<\displaystyle\frac{k_{p}^{2}}{\epsilon_{0}}$ so that real part of
$\epsilon$ is negative and hence we can have solution of the above dispersion
equations. This gives us another sequence of resonances
$k^{2}_{n}:=\displaystyle\frac{k^{2}_{p}}{\epsilon_{0}}\left(\epsilon_{0}-\frac{1}{2}-\sigma_{n}\right)$
(i.e. corresponding to the sequence of plasmonics in electromagnetism).

  4. (3)

Third situation

(1.8) $a:=\begin{cases}a_{j}\sim 1\quad\text{inside }D_{j},\\\
a_{0}\quad\text{outside
}\overset{M}{\underset{j=1}{\cup}}D_{j},\end{cases}\text{and}\qquad
b:=\begin{cases}b_{j}\sim\delta^{-2}\quad\text{inside }D_{j},\\\
b_{0}\quad\text{outside }\overset{M}{\underset{j=1}{\cup}}D_{j}.\end{cases}$

In this case, we have $\alpha\sim 1$ and then we keep away from the full
spectrum of $(K_{D}^{0})^{*}$. Hence, there is no singularity coming from
(1.5). But as $\gamma\sim\delta^{-2}\gg 1$, we can excite the eigenvalues of
the Newtonian operators $A^{0}$ and create singularities in (1.4). This gives
us a sequence of resonances with volumetric-modes. Indeed, we can write, after
scaling,

$I-\gamma k^{2}A^{0}=I-\gamma k^{2}\,\delta^{2}\,\widetilde{A^{0}},$

where

$\widetilde{A^{0}}\;v(x):=\int_{B}\,\Phi_{0}(|x-y|)\,v(y)\,dy.$

Let $\\{\tilde{\lambda}_{n}\\}_{n\geq 1},$ be the sequence of eigenvalues of
$\widetilde{A^{0}}$. Then, the operator $[I-\gamma k^{2}A^{0}]$ is not
injective if $k$ satisfies one of the following dispersion equations $1-\gamma
k^{2}\delta^{2}\tilde{\lambda}_{n}=0$. The values
$k_{n}:=\sqrt{\displaystyle\frac{1}{\gamma\,\delta^{2}\,\tilde{\lambda}_{n}}}$
are called the dielectric resonances. This was observed in [5] and [20].

The scattered field by the collection of inhomogeneities $D_{j},j=1,...,M,$ is
given by a linear combination of poles or dipoles centered at the centers of
the inhomogeneities. We call this dominating field the Foldy-Lax field as it
is of the form of the scattered field generated by point-like scatterers (or
Dirac-like potentials). This field encodes the multiples scattering effect
between the small inhomogeneities. A natural, and actually an outstanding open
question, was whether from the scattered field measured away from the
collection of inhomogeneities we can reconstruct the field due to multiple
interaction between them. In mathematical terms, is the approximating term,
i.e. the Foldy-Lax field or its iterations (see later), dominating the error
of the approximation? We call this property, the Foldy-Lax paradigm (or
approximation). There are attempts to justify this paradigm but without real
success. To our knowledge, the only published paper addressing this issue is
[13] where acoustic waves in the harmonic regime was considered with sound-
soft inhomogeneities (obstacles with Dirichlet condition). The Foldy-Lax
approximation was justified when the $D_{j}$’s are discs. The particular
geometry of these inhomogeneities allow them to perform exact computations.

In this work, we show that for general shapes and even general transmission
condition the Foldy-Lax approximation is valid as soon as nearly resonating
incident frequencies are used. Our message is that the main reason why we can
see the multiple scattering between the inhomogeneities is the fact that we
excite them with incident frequencies close to their own resonances. To
justify this claim, we consider inhomogeneities satisfying the conditions in
(1.8), assuming that $a_{j}=a_{0}$ as well, and take $k$ close to the
dielectric resonances $k_{n}:=\sqrt{1/\gamma\,\delta^{2}\,\lambda_{n}}$.
However, we do believe that similar results can be shown using the other
condition (1.6) and (1.7) and use incident frequencies close to the Minnaert
resonance
$k_{M}:=\sqrt{\displaystyle\frac{8\,\pi\,a_{0}}{a_{1}\,\Theta_{\partial D}}}$
or the plasmonic resonances
$\displaystyle\frac{k^{2}_{p}}{\epsilon_{0}}\left(\epsilon_{0}-\frac{1}{2}-\sigma_{n}\right)$.

To state mathematically these results, we first set $d:=\underset{i\neq j\atop
i,j=1;\cdots;M}{\min}\;dist(D_{i},D_{j})$ and assume this minimum distance to
be of the order:

(1.9) $d=d_{0}\;\delta^{t},\;t\geq 0,$

in the $3$ dimension case and

(1.10) $d=d_{0}\;e^{-|log(\delta)|^{t}},\;t\geq 0,$

in the $2$ dimension case. In both cases, the constant $d_{0}>0$ is
independent on $\delta$.

In the next theorem, we discuss the Foldy-Lax approximation when the
inhomogeneities satisfy (1.8) in the $3$ dimension case. In this case, we
recall that the Newtonian operator $A^{0}:=A^{0}_{j}$, on any $D_{j}$, reads

(1.11)
$A^{0}:\mathbb{L}^{2}(D_{j})\longrightarrow\mathbb{L}^{2}(D_{j}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
A^{0}(u)(x):=\int_{D_{j}}\Phi_{0}(x,y)u(y)dy=\int_{D_{j}}\frac{1}{4\pi|x-y|}\;u(y)dy.$

######  Theorem 1.1.

Let $k_{n}:=\sqrt{\displaystyle\frac{1}{\gamma\,\lambda_{n}}}$ where
$\lambda_{n}$ is any eigenvalue of the Newtonian operator $A^{0}$ 222Recall
that in 3D, $\lambda_{n}=\delta^{2}\tilde{\lambda}_{n}$ where
$\tilde{\lambda}_{n}$ are the eignevalues of $\widetilde{A_{0}}$.. We choose
the incident frequency $k$ satisfying

(1.12) $k^{2}:=k^{2}_{n}\left(1\pm\delta^{h}\right),\;\delta<<1.$

Under the condition $t+h\leq 1$, we have the expansion:

(1.13)
$u^{s}(x,\theta)=\sum^{M}_{j=1}\Phi_{k}(x,z_{j})\;C_{j}\;Q_{j}\;+\mathcal{O}\left(\delta^{\min\\{2-h,3-2h-2t\\}}\right),$

where $Q:=(Q_{1},...,Q_{M})$ is the unique solution of the algebraic system:

$\left(I-B_{k}\right)\cdot Q\;=\;U,$

where $U:=(u^{i}(z_{1},\theta),\cdots,u^{i}(z_{M},\theta))$ and
$B_{k}:=\left(B_{k,i,j}\right)^{M}_{i,j=1}$ with333Where $\bm{\delta}$ is the
Kronecker symbol.

$\displaystyle B_{k,i,j}$ $\displaystyle:=$
$\displaystyle\frac{C_{j}}{\left(1-i\,k\,C_{i}\right)}\,\,\Phi_{k}(z_{i},z_{j})\,\left(1-\underset{i,j}{\bm{\delta}}\right)$
and $\displaystyle C_{j}$ $\displaystyle:=$
$\displaystyle\int_{D_{j}}\left[\frac{a_{0}}{k^{2}\,b}I-A^{0}\right]^{-1}(1)(x)\,dx.$

As a consequence of the previous theorem, we state the following corollary.

######  Corollary 1.2.

For any integer $N$, we define:

$Q^{N}:=\sum_{n=0}^{N}\,B_{k}^{n}\cdot U\quad\text{and}\quad
u^{s,N}(x,\theta):=\sum^{M}_{j=1}\Phi_{k}(x,z_{j})\;C_{j}\;Q^{N}_{j}.$

  1. (1)

Under the following condition:

(1.14) $0\leq 1-h-t\leq\min\left\\{\frac{1}{N+1},\frac{1-t}{N}\right\\},$

we have

(1.15)
$u^{s}(x,\theta)-u^{s,N}(x,\theta)=\mathcal{O}\left(\delta^{(1-h)+(N+1)(1-t-h)}\right)$

and

(1.16)
$u^{s,N}(x,\theta)-u^{s,N-1}(x,\theta)\;\sim\;\delta^{(1-h)+N(1-t-h)}\gg\delta^{(1-h)+(N+1)(1-t-h)},$

for any bounded domain away from the collection of centers $z_{j},j=1,...,M$.
Here, $u^{s,N}(x,\theta)$ is the scattered field after $N$-interactions
between the nano-particles.

  2. (2)

Under the limit condition $1-t-h=0$ and $C_{0}\,d_{0}<1$, where $C_{0}$ is
such that $C=C_{0}\,\delta^{1-h}$ and $d_{0}$ satisfy $(\ref{d03D})$, we have
the expansion:

(1.17)
$u^{s}(x,\theta)-u^{s,\infty}(x,\theta)=\mathcal{O}\left(\delta\right),\;\delta\ll
1,$

where

$u^{s,\infty}(x,\theta):=\sum^{M}_{j=1}\Phi_{k}(x,z_{j})\;C_{j}\;Q^{\infty}_{j}\sim\delta^{1-h}\sim\delta^{t}$

for in any bounded domain away from the collection of centers
$z_{j},j=1,\cdots,M$. Here $u^{s,\infty}(x,\theta)$ is the field generated
after all the interactions between the particles, i.e.
$Q^{\infty}:=(Q_{1}^{\infty},\cdots,Q_{M}^{\infty})$ is given by

(1.18) $Q^{\infty}=\sum^{\infty}_{n=0}B^{n}_{k}\cdot U.$

We call $u^{s,\infty}(\cdot,\theta)$ the Foldy-Lax field.

The part (1) of Corollary 1.2 means that if we want to reconstruct the field
encoding the interactions between the particles until the order $N$, for a
given integer $N$, we need to use an incident frequency close to the resonance
at the order $O(a^{h})$ with $h$ satisfying (1.14). In addition, under the
optimal condition $1-t-h=0$, we can see all the interactions between the
particles. Therefore under this condition between the closeness of the
particles and closeness of the incident frequency to the used chosen
resonance, the Foldy field can be fully reconstructed and hence the whole
interactions between the particles can be ’seen’.

The coming corollary, gives more details about the case when the particles are
away from each other, i.e. when we can take $t=0$.

######  Corollary 1.3.

In this case, the condition (1.14), reads as:

(1.19) $0\leq 1-h\leq\frac{1}{N+1}.$

and

(1.20)
$u^{s}(x,\theta)-u^{s,N}(x,\theta)=\mathcal{O}\left(\delta^{(N+2)(1-h)}\right),$

with

$u^{s,N}(x,\theta)-u^{s,N-1}(x,\theta)\sim\delta^{(N+1)(1-h)}\gg\delta^{(N+2)(1-h)}.$

The estimate (1.20) means that the Foldy-Lax approximation, at a given order
of interaction $N$, is also valid when the nano-particles are away from each
other as soon as the used incident frequency is close to the resonance
$k_{n_{0}}$, i.e. (1.12) is valid with $h\in(0,1)$ such that (1.19) is
satisfied.

Next, we state the corresponding results for the $2$ dimensional case. For
this, we state the natural Newtonian operator $A^{0}:=A^{0}_{j}$ as:

(1.21)
$A^{0}:\mathbb{L}^{2}(D_{j})\longrightarrow\mathbb{L}^{2}(D_{j}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
A^{0}(u)(x):=\int_{D_{j}}\Phi_{0}(x,y)u(y)dy=-\int_{D_{j}}\frac{1}{2\pi}\log(|x-y|)\;u(y)dy.$

In dimension 2, we need the following contrast of the coefficients $a_{j}$ and
$b_{j}$.

(1.22) $a:=\begin{cases}a_{j}\sim 1\quad\text{inside }D_{j},\\\
a_{0}\quad\text{outside
}\overset{M}{\underset{j=1}{\cup}}D_{j},\end{cases}\text{and}\qquad
b:=\begin{cases}b_{j}\sim\delta^{-2}\left|\log(\delta)\right|^{-1}\quad\text{inside
}D_{j},\\\ b_{0}\quad\text{outside
}\overset{M}{\underset{j=1}{\cup}}D_{j}.\end{cases}$

Compared to the 3D case, here we have contrasts of the order
$\delta^{-2}\left|\log(\delta)\right|^{-1}$ instead of $\delta^{-2}$. Such
scales are dictated by the $\log$-type singularity of the fundamental
solution, see [17]. Therefore, we need the following hypothesis on the
behavior of the eigenvalues and eigenfunctions of the 2D Newtonian operator:

######  Hypotheses 1.

The particles $D_{j}$, of radius $\delta,\;\delta\ll 1$, are taken such that
the spectral problem $A_{0}u=\lambda\;u,\mbox{ in }D_{j}$, has eigenvalues
$\lambda_{n}$ and corresponding eigenfunctions, $e_{n}$, satisfying the
following properties:

  1. (1)

$\int_{D_{j}}e_{n}(x)dx\neq 0,\;\forall\delta\ll 1.$

  2. (2)

$\lambda_{n}\sim\delta^{2}|\log(\delta)|,\;\forall\delta\ll 1.$

In [17], see the appendix there, it is shown that for particles of general
shapes, the first eigenvalue and the corresponding eigenfunctions satisfy
Hypotheses 1. In addition, the properties of the eigenvalues for the case when
the particles are discs are characterized.

######  Theorem 1.4.

Assume Hypotheses 1 to be satisfied and let
$k_{n}:=\sqrt{\displaystyle\frac{1}{\gamma\;\lambda_{n}}}$ where $\lambda_{n}$
is any eigenvalue of the Logarithmic Newtonian operator $A^{0}$ 444From
(1.22), we see that $\gamma\sim\delta^{-2}|\log(\delta)|^{-1}$. Therefore and
the second part of Hypotheses 1, we deduce that $k_{n}=O(1)$ for $\delta\ll
1$.. We choose the incident frequency $k$ satisfying

(1.23)
$k^{2}:=k^{2}_{n}\left(1\pm\left|\log(\delta)\right|^{-h}\right),\;\;\delta<<1.$

Under the condition $t+h\leq 1$, we have the expansion:

$\displaystyle u^{s}(x,\theta)$ $\displaystyle=$
$\displaystyle\sum^{M}_{j=1}\Phi_{k}\left(x,z_{j}\right)\;C_{j}^{\star}\;Q_{j}+\mathcal{O}\left(\delta^{1-t}\,\left|\log(\delta)\right|^{2\,(h-1)}\right)+\mathcal{O}\left(\delta\,\left|\log(\delta)\right|^{(h-1)}\right),$

where $Q:=\left(Q_{1},\cdots,Q_{M}\right)$ is the unique solution of the
algebraic system

$\left(I-B_{k}\right)\cdot Q\;=\;U,$

where $U:=\left(u^{i}(z_{1},\theta),\cdots,u^{i}(z_{M},\theta)\right)$ and
$B_{k}:=(B_{k,i,j})^{M}_{i,j=1}$ where

(1.24)
$B_{k,i,j}:=\Phi_{k}\left(z_{i},z_{j}\right)\;\left[C^{-1}_{j}\,-\,\bm{E}\right]^{-1}\;\left(1-\underset{i,j}{\bm{\delta}}\right)$

and

$\displaystyle C_{j}$ $\displaystyle:=$
$\displaystyle\int_{D_{j}}\left[\frac{a_{0}}{k^{2}\,b}I-A^{0}\right]^{-1}(1)(x)dx$

with

$\bm{E}:=\frac{i}{4}-\frac{1}{2\pi}\left[\log\left(\frac{k}{2}\right)+\displaystyle\lim_{p\rightarrow+\infty}\left(\sum_{m=1}^{p}\frac{1}{m}-\log(p)\right)\right].$

The previous theorem suggest the following corollary.

######  Corollary 1.5.

For any integer $N$, we define

$Q^{N}:=\sum_{n=0}^{N}\,B_{k}^{n}\cdot U\quad\text{and}\quad
u^{s,N}(x,\theta):=\sum^{M}_{j=1}\Phi_{k}(x,z_{j})\;\left[C_{j}^{-1}\,-\,\bm{E}\right]^{-1}\;Q^{N}_{j}.$

  1. (1)

Under the following general condition, which is uniform in terms of
$N<\infty$,

(1.25) $1-t-h>0$

we have

(1.26)
$u^{s}(x,\theta)-u^{s,N}(x,\theta)=\mathcal{O}\left(\left|\log(\delta)\right|^{(h-1)-(N+1)(1-t-h)}\right)$

and

(1.27)
$u^{s,N}(x,\theta)-u^{s,N-1}(x,\theta)\sim\left|\log(\delta)\right|^{(h-1)-N(1-t-h)}\gg\left|\log(\delta)\right|^{(h-1)-(N+1)(1-t-h)}$

for in any bounded domain away from the collection of centers
$z_{j},j=1,\cdots,M$.

  2. (2)

Under the limit condition $1-t-h=0$, we have the expansion

(1.28)
$u^{s}(x,\theta)-u^{s,\infty}(x,\theta)=\mathcal{O}\left(\delta^{1-t}\;\left|\log(\delta)\right|^{-2\,t}\right),\;\delta\ll
1,$

where

$u^{s,\infty}(x,\theta):=\sum^{M}_{j=1}\Phi_{k}(x,z_{j})\;\left[C_{j}^{-1}-\bm{E}\right]^{-1}\;Q^{\infty}_{j}\sim\left|\log(\delta)\right|^{-t}$

for in any bounded domain away from the collection of centers
$z_{j},j=1,\cdots,M$. Here $u^{s,\infty}(x,\theta)$ is the field generated
after all the interactions between the particles, i.e.
$Q^{\infty}:=(Q_{1}^{\infty},\cdots,Q_{M}^{\infty})$ is given by

$Q^{\infty}=\sum^{\infty}_{n=0}B_{k}^{n}\cdot U.$

We observe that, contrary to the $3$ dimensional case, the scattered field
generated after $N$ interactions (for any order $N$), can be reconstructed
from measured scattered field away from the cluster of particles. In addition,
the whole Foldy-Lax field can be reconstructed as well. Observe also that, we
can take $t=0$ (and then $h=1$), which means that the particles can be far
away from each other and still we have the Foldy-Lax approximation (for
$N\leq\infty$) if the incident frequency is close to the resonance with the
(optimal value) $h=1$. This means that in the $2D$ case, the Foldy-Lax
approximation is always valid. In the case $N<\infty$, we can also take $h=0$
in (1.25) and hence we do not need to use nearly resonating incident
frequencies (this is not the case for $N=\infty$). As in (1.25), $h=0$ implies
that $t=1$, therefore the Foldy-Lax approximation is valid but in the
mesoscale regime, see (1.10), i.e. $d\sim\delta$, if the incident frequency is
not close to any resonance.
At the mathematical analysis level, in the $2$ D case, the singularity of the
fundamental solution is of log-type and hence its powers stay always
dominating the singularity of its derivatives. This is not true in the $3$ D
case.
The results in Theorem 1.4 confirm, in particular, the ones in [13] where in
the 2D case, it is shown that the Foldy-Lax approximation is valid for sound-
soft acoustic obstacles of discs shapes.

Let us finish this introduction by mentioning that the Foldy-Lax
approximations have different applications, see [19] for classical results and
historical facts. This approximation is called after Foldy and Lax as the
dominant part of these approximation is nothing but the field generated by
point-like inhomogeneity as modeled by Foldy [16] and Lax [18]. In addition,
such approximations are useful in modern mathematical imaging see for instance
[2, 1, 17] and the effective medium theory [14, 3].

##  2\. Proof of Theorem 1.1

We start the proof by setting the Lippmann-Schwinger equation, for several
particles, given for $x\in D_{m}$ by:

$v_{m}(x)-k^{2}\,\frac{1}{a_{0}}\,\tau\int_{D_{m}}\Phi_{k}(x,y)v_{m}(y)dy-k^{2}\,\frac{1}{a_{0}}\,\tau\sum_{j\neq
m}\int_{D_{j}}\Phi_{k}(x,y)v_{j}(y)dy=u^{i}(x),$

where we recall that555To avoid more complicated notations and simplify the
exposition, we assume that $\tau_{j}:=b-b_{0}(z_{j})=\tau,\,\,j=1,\cdots,M.$
The final results stay valid without this assumption.
$\tau:=b-b_{0}(z)\sim\delta^{-2}$ and $v_{m}(\cdot)=u(\cdot)_{|_{D_{m}}}$. To
make appear the Newtonian operator, we rewrite the previous equation as:

$\displaystyle v_{m}(x)$ $\displaystyle-$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\tau\,\int_{D_{m}}\Phi_{0}(x,y)\,v_{m}(y)\,dy-\,k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{j\neq
m}\int_{D_{j}}\Phi_{k}(x,y)\,v_{j}(y)\,dy$ $\displaystyle=$ $\displaystyle
u^{i}(x)+k^{2}\,\frac{1}{a_{0}}\,\tau\,\int_{D_{m}}(\Phi_{k}-\Phi_{0})(x,y)\,v_{m}(y)\,dy.$

By expanding, using Taylor expansion, the functions $\Phi_{k}(\cdot,\cdot)$
and $u^{i}(\cdot)$ near the centers and using the series representation of
$(\Phi_{k}-\Phi_{0})$, we obtain:

$\displaystyle\left[I-k^{2}\,\frac{1}{a_{0}}\,\tau\,A^{0}\right]v_{m}(x)$
$\displaystyle-$ $\displaystyle\,k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{j\neq
m}\Phi_{k}(z_{m},z_{j})\,\int_{D_{j}}\,v_{j}(y)\,dy$ $\displaystyle-$
$\displaystyle\,k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{j\neq
m}\int_{D_{j}}\,\int_{0}^{1}\underset{x}{\nabla}\Phi_{k}(z_{m}+t(x-z_{m}),z_{j})\cdot(x-z_{m})\,dt\,v_{j}(y)\,dy$
$\displaystyle-$ $\displaystyle\,k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{j\neq
m}\int_{D_{j}}\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x,z_{j}+t(y-z_{j}))\cdot(y-z_{j})\,dt\,v_{j}(y)\,dy$
$\displaystyle=$ $\displaystyle u^{i}(z_{m})+\int_{0}^{1}\nabla
u^{i}(z_{m}+t(x-z_{m}))\,\cdot(x-z_{m})\,dt$ $\displaystyle+$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\tau\,\int_{D_{m}}\left[ik+\sum_{\ell\geq
1}\frac{(i\,k)^{\ell+1}}{(\ell+1)!}\,\left|x-y\right|^{\ell}\right]\,v_{m}(y)\,dy.$

We set the scattering coefficient $w$ to be:

$w:=k^{2}\,\frac{1}{a_{0}}\,\tau\,\left[I-k^{2}\,\frac{1}{a_{0}}\,\tau\,A^{0}\right]^{-1}(1).$

On both sides of the previous equation, we take the inverse operator of
$\left[I-k^{2}\,\frac{1}{a_{0}}\,\tau\,A^{0}\right]$ and then integrating over
$D_{m}$ to obtain:

(2.1)
$\displaystyle\left[1-ik\int_{D_{m}}w(x)\,dx\right]\,\int_{D_{m}}v_{m}(x)\,dx$
$\displaystyle=$
$\displaystyle\frac{a_{0}}{k^{2}\,\tau}u^{i}(z_{m})\int_{D_{m}}w(x)\,dx$
$\displaystyle+$ $\displaystyle\sum_{j\neq
m}\Phi_{k}(z_{m},z_{j})\int_{D_{j}}v_{j}(y)\,dy\,\int_{D_{m}}w(x)\,dx$
$\displaystyle+$ $\displaystyle Remainder,$

where the remainder term is defined by:

(2.2) $\displaystyle Remainder$ $\displaystyle:=$
$\displaystyle\frac{a_{0}}{k^{2}\,\tau}\,\int_{D_{m}}w(x)\,\int_{0}^{1}\nabla
u^{i}(z_{m}+t(x-z_{m}))\centerdot(x-z_{m})\,dt\,dx$ $\displaystyle+$
$\displaystyle\int_{D_{m}}w(x)\int_{D_{m}}\sum_{\ell\geq
1}\frac{(ik)^{\ell+1}}{(\ell+1)!}\,\left|x-y\right|^{\ell}v_{m}(y)\,dy\,dx$
$\displaystyle+$ $\displaystyle\int_{D_{m}}w(x)\sum_{j\neq
m}\int_{D_{j}}\,\int_{0}^{1}\underset{x}{\nabla}\Phi_{k}(z_{m}+t(x-z_{m});z_{j})\cdot(x-z_{m})\,dt\,v_{j}(y)\;dy\;dx$
$\displaystyle+$ $\displaystyle\int_{D_{m}}w(x)\sum_{j\neq
m}\int_{D_{j}}\,\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x;z_{j}+t(y-z_{j}))\cdot(y-z_{j})\,dt\,v_{j}(y)\;dy\;dx.$

In the sequel, we will derive an estimation for $Remainder$. For this, we need
the following proposition.

######  Proposition 2.1.

The total field, in the presence of particles, can be estimated by the source
field via the following inequality:

(2.3) $\|v\|\leq\delta^{-h}\,\|u\|.$

The scattering coefficient satisfies the following estimations:

(2.4)
$C_{m}:=\int_{D_{m}}w(x)\,dx=\mathcal{O}\left(\delta^{1-h}\right)\quad\text{and}\quad\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}=\mathcal{O}\left(\delta^{-\frac{1}{2}-h}\right).$

######  Proof.

For the proof of this proposition, we refer the readers to Appendix 4.
$\square$

We split $(\ref{Remainder})$ into four terms that can be estimated separately.

  1. $\ast$

Estimation of:

$\displaystyle T_{1}$ $\displaystyle:=$
$\displaystyle\frac{a_{0}}{k^{2}\,\tau}\,\int_{D_{m}}w(x)\,\int_{0}^{1}\nabla
u^{i}(z_{m}+t(x-z_{m}))\centerdot(x-z_{m})\,dt\,dx$ (2.5)
$\displaystyle\left|T_{1}\right|$ $\displaystyle\lesssim$
$\displaystyle\tau^{-1}\,\left|\int_{D_{m}}w(x)\,\int_{0}^{1}\nabla
u^{i}(z_{m}+t(x-z_{m}))\centerdot(x-z_{m})\,dt\,dx\right|$
$\displaystyle\lesssim$
$\displaystyle\tau^{-1}\,\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\left\|\int_{0}^{1}\nabla
u^{i}(z_{m}+t(\cdot-z_{m}))\centerdot(\cdot-
z_{m})\,dt\right\|_{\mathbb{L}^{2}(D_{m})}$ $\displaystyle=$
$\displaystyle\mathcal{O}\left(\delta^{\frac{9}{2}}\,\;\parallel
w\parallel_{\mathbb{L}^{2}(D_{m})}\right),$

where the last estimation can be justified using the smoothness of the
incident field, $u^{i}(\cdot)$, and the fact that $\tau\sim\delta^{-2}$.

  2.   3. $\ast$

Estimation of:

$\displaystyle T_{2}$ $\displaystyle:=$
$\displaystyle\int_{D_{m}}w(x)\int_{D_{m}}\sum_{\ell\geq
1}\frac{(ik)^{\ell+1}}{(\ell+1)!}\,\left|x-y\right|^{\ell}v_{m}(y)\,dy\,dx$
(2.6) $\displaystyle\left|T_{2}\right|$ $\displaystyle\leq$
$\displaystyle\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\,\left\|\int_{D_{m}}\sum_{\ell\geq
1}\frac{(ik)^{\ell+1}}{(\ell+1)!}\,\left|\cdot-y\right|^{\ell}v_{m}(y)\,dy\right\|_{\mathbb{L}^{2}(D_{m})}$
$\displaystyle=$
$\displaystyle\mathcal{O}\left(\delta^{4}\,\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\left\|v_{m}\right\|_{\mathbb{L}^{2}(D_{m})}\right)\overset{(\ref{aprioriestimation3D})}{=}\mathcal{O}\left(\delta^{\frac{11}{2}-h}\,\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\right).$

  4.   5. $\ast$

Estimation of:

$\displaystyle T_{3}$ $\displaystyle:=$
$\displaystyle\int_{D_{m}}w(x)\sum_{j\neq
m}\int_{D_{j}}\,\int_{0}^{1}\underset{x}{\nabla}\Phi_{k}(z_{m}+t(x-z_{m});z_{j})\cdot(x-z_{m})\,dt\,v_{j}(y)\;dy\;dx$
(2.7) $\displaystyle\left|T_{3}\right|$ $\displaystyle\leq$
$\displaystyle\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\sum_{j\neq
m}\left\|\int_{D_{j}}\,\int_{0}^{1}\underset{x}{\nabla}\Phi_{k}(z_{m}+t(\cdot-
z_{m});z_{j})\cdot(\cdot-
z_{m})\,dt\,v_{j}(y)\;dy\right\|_{\mathbb{L}^{2}(D_{m})}$
$\displaystyle\lesssim$
$\displaystyle\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\delta^{4}\,\sum_{j\neq
m}\frac{1}{\left|z_{m}-z_{j}\right|^{2}}\,\,\left\|v_{j}\right\|_{\mathbb{L}^{2}(D_{m})}$
$\displaystyle\leq$
$\displaystyle\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\delta^{4}\,\left[\sum_{j\neq
m}\frac{1}{\left|z_{m}-z_{j}\right|^{4}}\right]^{\frac{1}{2}}\,\,\left[\sum_{j\neq
m}\left\|v_{j}\right\|^{2}_{\mathbb{L}^{2}(D_{m})}\right]^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\delta^{4}\,d^{-2}\,\,\left\|v\right\|_{\mathbb{L}^{2}(D)}\overset{(\ref{aprioriestimation3D})}{=}\mathcal{O}\left(\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\delta^{\frac{11}{2}-2t-h}\right).$

  6.   7. $\ast$

Estimation of:

$T_{4}:=\int_{D_{m}}w(x)\sum_{j\neq
m}\int_{D_{j}}\,\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x;z_{j}+t(y-z_{j}))\cdot(y-z_{j})\,dt\,v_{j}(y)\;dy\;dx.$

Because the expression of $T_{4}$ is similar to the one of $T_{3}$ and to
avoid the redundancy of computations, we skip the computation steps for its
estimation and we deduce that:

(2.8)
$T_{4}=\mathcal{O}\left(\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\delta^{4}\,d^{-2}\,\,\left\|v\right\|_{\mathbb{L}^{2}(D)}\right)\overset{(\ref{aprioriestimation3D})}{=}\mathcal{O}\left(\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\delta^{\frac{11}{2}-2t-h}\right).$

Then, by gathering $(\ref{estimationT1})-(\ref{estimationT4})$, the remainder
term given by $(\ref{Remainder})$ will be estimated as:

(2.9)
$Remainder=\mathcal{O}\left(\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\delta^{\min(\frac{11}{2}-2t-h;\frac{9}{2})}\right)\overset{(\ref{estimationnormw})}{=}\mathcal{O}\left(\delta^{4-h+\min(1-2t-h;0)}\right).$

Finally, using the estimation $(\ref{estimationRemainder})$, the algebraic
system given by $(\ref{AlgebraicSystem})$ becomes:

$\displaystyle\left[1-i\,k\,\int_{D_{m}}w(x)\,dx\right]\;\int_{D_{m}}v_{m}(x)dx$
$\displaystyle=$
$\displaystyle\frac{a_{0}}{k^{2}\,\tau}u^{i}(z_{m})\int_{D_{m}}w(x)\,dx$
$\displaystyle+$ $\displaystyle\int_{D_{m}}w(x)\,dx\,\sum_{j\neq
m}\Phi_{k}(z_{m},z_{j})\int_{D_{j}}v_{j}(y)\,dy$ $\displaystyle+$
$\displaystyle\mathcal{O}\left(\delta^{4-h+\min(1-2t-h;0)}\right).$

To write short the previous system we set, for $j=1,\cdots,M$, the following
notations:

$C_{j}:=\int_{D_{j}}w(x)\,dx,\,\,Q_{j}:=k^{2}\,\frac{1}{a_{0}}\,\tau\,C_{j}^{-1}\,\int_{D_{j}}v_{j}(x)\;dx\quad\text{and}\quad
U^{i}(z_{j}):=\left[1-\,i\,k\,C_{j}\right]^{-1}u^{i}(z_{j}).$

Then, the previous system will be reduced to:

(2.10) $Q_{m}-\left(1-i\,k\,C_{m}\right)^{-1}\,\sum_{j\neq
m}\Phi_{k}(z_{m};z_{j})\,C_{j}\,Q_{j}\,=U^{i}(z_{m})+\,Error_{m},$

where

(2.11) $Error_{m}=\mathcal{O}\left(\delta^{\min(2-2t-h;1)}\right).$

The matrix representation of the last algebraic system, see $(\ref{SA3D})$, is
as follows:

(2.12)
$\left(I-B_{k}\right)\;\overline{Q}\;=\;\overline{U}\;+\;\overline{Err},$

where

$B_{k}:=\begin{pmatrix}0&\displaystyle\frac{C_{2}\Phi_{k}(z_{1};z_{2})}{(1-ikC_{1})}&\cdots&\displaystyle\frac{C_{M}\Phi_{k}(z_{1};z_{M})}{(1-ikC_{1})}\\\
\displaystyle\frac{C_{1}\Phi_{k}(z_{2};z_{1})}{(1-ikC_{2})}&0&\cdots&\displaystyle\frac{C_{M}\Phi_{k}(z_{2};z_{M})}{(1-ikC_{2})}\\\
\vdots&\vdots&\ddots&\vdots\\\
\displaystyle\frac{C_{1}\Phi_{k}(z_{M};z_{1})}{(1-ikC_{M})}&\displaystyle\frac{C_{2}\Phi_{k}(z_{M};z_{2})}{(1-ikC_{M})}&\cdots&0\end{pmatrix}$

and

$\overline{Q}:=\left(Q_{j}\right)_{j=1}^{M},\,\,\overline{U}:=\left(U^{i}(z_{j})\right)_{j=1}^{M}\quad\text{and}\quad\overline{Err}:=\left(Error_{j}\right)_{j=1}^{M}.$

To investigate the invertibility of $(\ref{sysper})$, we start first by
proving the invertibility of the corresponding non perturbed linear system,
given by:

(2.13) $\left(I-B_{k}\right)\cdot\tilde{Q}\;=\;\overline{U},$

and then to justify that the difference, in the norm sense, between the
solution of the perturbed and the unperturbed problem is at most of the same
order as $\overline{Err}$.

To invert the algebraic system using the Born series, we must check under what
conditions $\|B_{k}\|<1$.

$\|B_{k}\|:=\max_{i=1,\cdots,M}\left\\{\sum_{j\neq
i}|C_{j}|\;\frac{|\Phi_{k}(z_{i};z_{j})|}{|1-ikC_{i}|}\right\\}=\max_{i=1,\cdots,M}\left\\{|C_{j}|\,\sum_{j\neq
i}\frac{1}{|z_{i}-z_{j}|\,|1-ikC_{i}|}\right\\}.$

As we know, from $(\ref{C=intW})$, that
$C_{j}=\mathcal{O}\left(\delta^{1-h}\right),\;j=1,\cdots,M$, we approximate
the previous formula by:

(2.14) $\|B_{k}\|=\delta^{1-h}\;\sum_{j\neq
i}\;\frac{1}{|z_{i}-z_{j}|}=\mathcal{O}\left(\delta^{1-h}\;d^{-1}\right)=\mathcal{O}\left(\delta^{1-h-t}\right).$

Hence, we deduce that:

(2.15) $\|B_{k}\|<1\Leftrightarrow 1-h-t>0.$

Now for the critical case, i.e. $1-h-t=0$, we put the following assumptions:

$\sum_{j\neq
i}\;\frac{1}{|z_{i}-z_{j}|}=d_{0}\;\,d^{-1}=d_{0}\,\delta^{-t}\quad\text{and}\quad
C_{j}=C_{0}\;\,\delta^{1-h}.$

Under these assumptions and the condition $1-h-t=0$, we deduce that:

(2.16) $\|B_{k}\|<1\Leftrightarrow C_{0}\,d_{0}\,<1.$

We have justified that under the conditions $(\ref{constantlesssmall})$ or
$(\ref{constantless1})$, for the critical case, the invertibility of the
unperturbed system, see $(\ref{sys})$.

Now, under the conditions of invertibility, by subtracting $(\ref{sysper})$
from $(\ref{sys})$ we obtain:

$\overline{Q}-\tilde{Q}=\left(I-B_{k}\right)^{-1}\;\overline{Err},$

and by taking the Euclidian norm both side of the equality, we get:

(2.17)
$\left\|\overline{Q}-\tilde{Q}\right\|=\left\|\left(I-B_{k}\right)^{-1}\;\overline{Err}\right\|\lesssim\left\|\overline{Err}\right\|:=\left[\sum_{m=1}^{M}\left|Error_{m}\right|^{2}\right]^{\frac{1}{2}}\overset{(\ref{Errorm})}{=}\mathcal{O}\left(\delta^{\min(2-2t-h;1)}\right).$

To prove $(\ref{intro-usca})$ we will compute an approximation for the
scattered field. More precisely, we start by setting the following L.S.E:

$u(x)-k^{2}\,\frac{1}{a_{0}}\,\tau\,\int_{D}\Phi_{k}(x,y)\,u(y)\,dy=u^{i}(x),$

which can be rewritten, knowing that $u(\cdot)=u^{s}(\cdot)+u^{i}(\cdot)$ ,
as:

$u^{s}(x)=k^{2}\,\frac{1}{a_{0}}\,\tau\,\int_{D}\Phi_{k}(x,y)\,u(y)\,dy=k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{m=1}^{M}\,\int_{D_{m}}\Phi_{k}(x,y)\,v_{m}(y)\,dy,$

where $v_{m}(\cdot)=u(\cdot)_{|_{D_{m}}}$, and we have assumed that $x$ is
away from $D=\underset{m=1}{\overset{M}{\cup}}D_{m}$. Then, using Taylor
expansion for the function $\Phi_{k}(x,\cdot)$ we obtain:

$\displaystyle u^{s}(x,\theta)$ $\displaystyle=$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\tau\sum_{m=1}^{M}\Phi_{k}(x,z_{m})\;\int_{D_{m}}v_{m}(y)\,dy$
$\displaystyle+$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{m=1}^{M}\,\int_{D_{m}}\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x,z_{m}+t(y-z_{m}))\cdot\left(y-z_{m}\right)dt\,v_{m}(y)\,dy.$

The second term on the right hand side can be estimated as:

$\displaystyle I_{1}$ $\displaystyle:=$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{m=1}^{M}\,\int_{D_{m}}\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x,z_{m}+t(y-z_{m}))\cdot\left(y-z_{m}\right)dt\,v_{m}(y)\,dy$
$\displaystyle\left|I_{1}\right|$ $\displaystyle\lesssim$
$\displaystyle\delta^{-2}\,\sum_{m=1}^{M}\left|\int_{D_{m}}\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x,z_{m}+t(y-z_{m}))\cdot\left(y-z_{m}\right)dt\,v_{m}(y)\,dy\right|$
$\displaystyle\lesssim$
$\displaystyle\delta^{-2}\,\sum_{m=1}^{M}\left\|\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x,z_{m}+t(\cdot-
z_{m}))\cdot\left(\cdot-
z_{m}\right)dt\right\|_{\mathbb{L}^{2}(D_{m})}\,\left\|v_{m}\right\|_{\mathbb{L}^{2}(D_{m})}$
$\displaystyle\lesssim$
$\displaystyle\delta^{\frac{1}{2}}\,\sum_{m=1}^{M}\left\|v_{m}\right\|_{\mathbb{L}^{2}(D_{m})}\leq\delta^{\frac{1}{2}}\,M^{\frac{1}{2}}\,\left(\sum_{m=1}^{M}\left\|v_{m}\right\|^{2}_{\mathbb{L}^{2}(D_{m})}\right)^{\frac{1}{2}}=\mathcal{O}\left(\delta^{\frac{1}{2}}\,\left\|v\right\|_{\mathbb{L}^{2}(D)}\right).$

Then,

$I_{1}=\mathcal{O}\left(\delta^{\frac{1}{2}}\,\left\|v\right\|_{\mathbb{L}^{2}(D)}\right)\overset{(\ref{aprioriestimation3D})}{=}\mathcal{O}\left(\delta^{2-h}\,\right).$
$\displaystyle u^{s}(x,\theta)$ $\displaystyle=$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\tau\sum_{m=1}^{M}\Phi_{k}(x,z_{m})\;\int_{D_{m}}v_{m}(y)\,dy+\mathcal{O}\left(\delta^{2-h}\right)$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{M}\Phi_{k}(x,z_{m})\;C_{m}\;Q_{m}\,+\mathcal{O}\left(\delta^{2-h}\right)$
$\displaystyle=$
$\displaystyle\langle\Phi_{k}(x,z)\;C;\overline{Q}\rangle+\mathcal{O}\left(\delta^{2-h}\right)$
$\displaystyle=$
$\displaystyle\langle\Phi_{k}(x,z)\;C;\tilde{Q}\rangle+\langle\Phi_{k}(x,z)\;C;(\overline{Q}-\tilde{Q})\rangle+\mathcal{O}\left(\delta^{2-h}\right).$

For the second term on the right hand side, we have:

$\displaystyle I_{2}$ $\displaystyle:=$
$\displaystyle\langle\Phi_{k}(x,z)\;C;(\overline{Q}-\tilde{Q})\rangle$
$\displaystyle\left|I_{2}\right|$ $\displaystyle\leq$
$\displaystyle\left\|\Phi_{k}(x,z)\;C\right\|\;\left\|\overline{Q}-\tilde{Q}\right\|\overset{(\ref{unper-
per})}{\lesssim}\left\|C\right\|\;\delta^{\min(2-2t-h;1)}.$

Thanks to $(\ref{C=intW})$, which gives us an estimation for the scattering
coefficient $C_{m}\sim\delta^{1-h}$, we obtain:

$\left|I_{2}\right|\lesssim
M^{\frac{1}{2}}\,\delta^{1-h}\;\delta^{\min(2-2t-h;1)}=\mathcal{O}\left(\delta^{1-h+\min(2-2t-h;1)}\right).$

Then,

(2.18) $\displaystyle u^{s}(x,\theta)$ $\displaystyle=$
$\displaystyle\langle\Phi_{k}(x,z)\;C;\tilde{Q}\rangle+\mathcal{O}\left(\delta^{1-h+\min(2-2t-h;1)}\right)+\mathcal{O}\left(\delta^{2-h}\right)$
$\displaystyle=$
$\displaystyle\langle\Phi_{k}(x,z)\;C;\tilde{Q}\rangle+\mathcal{O}\left(\delta^{\min(3-2t-2h;2-h)}\right).$

This proves $(\ref{intro-usca})$. We recall from $(\ref{sysper})$, that we
have:

$\left(I-B_{k}\right)\;\overline{Q}\;=\;\overline{U}\;+\;\overline{Err}.$

Then:

$\overline{Q}=\left(I-B_{k}\right)^{-1}\overline{U}+\left(I-B_{k}\right)^{-1}\overline{Err}=\sum_{n\geq
0}B_{k}^{n}\;\overline{U}+\overline{Err}.$

We have:

$\|B_{k}^{n}\;\overline{U}\|\leq\|B_{k}\|^{n}\;\|\overline{U}\|\simeq\|B_{k}\|^{n}\overset{(\ref{normBk})}{=}\mathcal{O}\left(\delta^{n(1-h-t)}\right).$

Now, we define the truncated solution of the unperturbed problem
$\tilde{Q}^{N}$, $N\in\mathbb{N}$, by:

$\tilde{Q}^{N}:=\sum_{n=0}^{N}\,B_{k}^{n}\cdot U$

and its associated scattered field

(2.19) $u^{s,N}(x,\theta):=\langle\Phi_{k}(x,z)\;C;\tilde{Q}^{N}\rangle.$

Subtracting $(\ref{TSF})$ from $(\ref{SF})$, we obtain:

$\displaystyle u^{s}(x,\theta)-u^{s,N}(x,\theta)$ $\displaystyle=$
$\displaystyle\langle\Phi_{k}(x,z)\;C;\tilde{Q}-\tilde{Q}^{N}\rangle+\mathcal{O}\left(\delta^{\min(3-2t-2h;2-h)}\right)$
(2.20) $\displaystyle\left|u^{s}(x,\theta)-u^{s,N}(x,\theta)\right|$
$\displaystyle=$ $\displaystyle\left|\langle\Phi_{k}(x,z)\;C;\sum_{n\geq
N+1}\,B_{k}^{n}\cdot
U\rangle\right|+\mathcal{O}\left(\delta^{\min(3-2t-2h;2-h)}\right)$
$\displaystyle\lesssim$
$\displaystyle\frac{|C|}{\underset{j=1,\cdots,M}{\min}|x-z_{j}|}\
\;\sum_{n\geq
N+1}\,\delta^{n(1-t-h)}+\mathcal{O}\left(\delta^{\min(3-2t-2h;2-h)}\right)$
$\displaystyle=$
$\displaystyle\delta^{1-h}\;\delta^{(N+1)(1-t-h)}+\mathcal{O}\left(\delta^{\min(3-2t-2h;2-h)}\right).$

We choose $h$ such that

$(1-h)+(N+1)(1-h-t)\leq\min\\{2-h,\;3-2h-2t\\},$

which can be rewritten as

(2.21) $0\leq 1-h-t\leq\min\left\\{\frac{1}{N+1};\frac{1-t}{N}\right\\},$

hence, $(\ref{219})$ becomes,

(2.22)
$u^{s}(x,\theta)-u^{s,N}(x,\theta)=\mathcal{O}\left(\delta^{(1-h)+(N+1)(1-t-h)}\right).$

This justifies $(\ref{intro-N-interactions-proof})$. In addition,

(2.23) $\displaystyle u^{s,N}(x,\theta)-u^{s,N-1}(x,\theta)$
$\displaystyle\overset{(\ref{TSF})}{=}$
$\displaystyle\langle\Phi_{k}(x,z)\,C;B_{k}^{N}\cdot U\rangle$
$\displaystyle\sim$
$\displaystyle\delta^{(1-h)+N(1-t-h)}>>\delta^{(1-h)+(N+1)(1-t-h)}$

for in any bounded domain away from the collection of centers
$z_{j},j=1,\cdots,M$. Here $u^{s,N}(x,\theta)$ is the scattered field after
$N$-interactions between the particles.

Under the limit condition $1-t-h=0$, the expansion in $(\ref{SF})$ is reduced
to:

$u^{s}(x,\theta)=\langle\Phi_{k}(x,z)\;C;\tilde{Q}\rangle+\mathcal{O}\left(\delta\right).$

If we set $u^{s,\infty}(x,\theta):=\langle\Phi_{k}(x,z)\;C;\tilde{Q}\rangle$,
then we obtain:

(2.24) $u^{s}(x,\theta)-u^{s,\infty}(x,\theta)=\mathcal{O}\left(\delta\right)$

which proves $(\ref{intro-infty-interactions-proof})$. Actually, from the
definition of $u^{s}(x,\theta)$, we have

$u^{s,\infty}(x,\theta):=\langle\Phi_{k}(x,z)\;C;\sum_{n=0}^{\infty}\,B_{k}^{n}\cdot
U\rangle=\sum_{n=0}^{\infty}\,\langle\Phi_{k}(x,z)\;C;B_{k}^{n}\cdot
U\rangle\sim\delta^{1-h}\sim\delta^{t}.$

is the field generated after all the interaction between the particles. We
call it the Foldy-Lax field. In addition, it is clear that for each term in
the previous series, i.e. any order $n\in\mathbb{N}$, we have
$\langle\Phi_{k}(x,z)\;C;B_{k}^{n}\cdot U\rangle\sim\delta^{1-h}>>\delta$, and
this allows us to justify that, at any order of interactions $n\in\mathbb{N}$,
at least one interaction between the $M$ particles can be seen from the
scattered field measured away from the cluster of particles. This ends the
proof of Corollary 1.2.

The justification of Corollary 1.3, can be handled in similar manner by taking
into account the fact that $t=0$.

##  3\. Proof of Theorem 1.4

We recall that the fundamental solution, $\Phi_{k}\left(\cdot,\cdot\right)$,
of the Helmholtz equation in dimension two satisfies

$\begin{cases}\underset{y}{\Delta}\Phi_{k}\left(x,y\right)+k^{2}\,\frac{b_{0}}{a_{0}}\,\Phi_{k}\left(x,y\right)=-\underset{x}{\delta}(y),\quad
x,y\in\mathbb{R}^{2},&\\\
\Phi_{k}\left(\cdot,\cdot\right)\;\;S.R.C\;\;\text{at}\;\;infinity,&\end{cases}$

where $k$ is such that $k^{2}=k^{2}\,\frac{b_{0}}{a_{0}}$ and it is given via
the Hankel function of first kind $H_{0}^{(1)}$. The following lemma, on the
decomposition of the Green’s kernel, is of importance to reduce the
complexities of the Lippmann-Schwinger equation, and then to invert it.

######  Lemma 3.1.

We have the following asymptotic expansion

(3.1)
$\Phi_{k}(x,y)=\Phi_{0}(x,y)+\bm{E}+\mathcal{O}(|x-y|^{2}\;\log(|x-y|)),\,\,x\,\,near\,\,y,$

where $\Phi_{0}(\cdot,\cdot)$ is the fundamental solution of the Laplace
equation in dimension two given by:

$\Phi_{0}(x,y):=\frac{-1}{2\,\pi}\,\log\left(\left|x-y\right|\right),\quad
x\neq y,$

and $\bm{E}$ is the constant given by:

$\bm{E}:=\frac{i}{4}-\frac{1}{2\pi}\left[\log\left(\frac{k}{2}\right)+\lim_{p\rightarrow+\infty}\left(\sum_{m=1}^{p}\frac{1}{m}-\log(p)\right)\right].$

###### Proof.

See formula $(3.84)$, page 74 in [15]. ∎

We start our proof of Theorem 1.4, by setting the following Lippmann-Schwinger
equation:

$v_{m}(x)-k^{2}\,\frac{1}{a_{0}}\,\tau\,\int_{D_{m}}\Phi_{k}(x,y)\,v_{m}(y)\,dy-k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{j\neq
m}\int_{D_{j}}\Phi_{k}(x,y)\,v_{j}(y)\,dy=u^{i}(x),\quad x\in D_{m},$

where, we assume that666As in 3D case, to simplify the computations, we write
the detailed proof with $\tau$, but of course we can do it with
$\left\\{\tau_{j}\right\\}_{j=1}^{M}$., $\tau_{j}=\tau,\;\;j=1,\cdots,M$ as
well. Then, thanks to Lemma 3.1, we rewrite the previous L.S.E as:

(3.2) $\displaystyle v_{m}(x)$ $\displaystyle-$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\tau\,\int_{D_{m}}\Phi_{0}(x,y)\,v_{m}(y)\,dy-k^{2}\,\frac{1}{a_{0}}\,\tau\,\bm{E}\,\int_{D_{m}}v_{m}(y)\,dy$
$\displaystyle-$ $\displaystyle k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{j\neq
m}\int_{D_{j}}\Phi_{k}(x,y)\,v_{j}(y)\,dy$ $\displaystyle=$ $\displaystyle
u^{i}(x)+\tau\,\int_{D_{m}}\mathcal{O}(|x-y|^{2}\;\log(|x-y|))\,v_{m}(y)\,dy.$

We introduce the Newtonian Potential operator $A^{0}$, defined by:

(3.3)
$A^{0}\left(v_{m}\right)(x)=\int_{D_{m}}\frac{-1}{2\pi}\log(|x-y|)\;v_{m}(y)\,dy.$

Then, going back to $(\ref{equa1})$, using the definition of the Newtonian
Potential operator, see $(\ref{Newpot2D})$, and expanding the Green’s kernel
and the incident field near the points $\left(z_{i}\right)_{i=1}^{M}$, we get:

(3.4)
$\displaystyle\left[I-k^{2}\,\frac{1}{a_{0}}\,b\,A^{0}\right]\left(v_{m}\right)(x)$
$\displaystyle-$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\;b\;\bm{E}\int_{D_{m}}v_{m}(y)\,dy$ $\displaystyle-$
$\displaystyle k^{2}\,\frac{1}{a_{0}}\,b\,\sum_{j\neq
m}\Phi_{k}(z_{m},z_{j})\,\int_{D_{j}}v_{j}(y)\,dy=u^{i}(z_{m})+Err_{1}(x),$

where

(3.5) $\displaystyle Err_{1}(x)$ $\displaystyle:=$
$\displaystyle\int_{0}^{1}\,\nabla u^{i}(z_{m}+t(x-z_{m}))\cdot(x-z_{m})\,dt$
$\displaystyle+$
$\displaystyle\tau\,\int_{D_{m}}\mathcal{O}(|x-y|^{2}\;\log(|x-y|))\,v_{m}(y)\,dy$
$\displaystyle+$ $\displaystyle k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{j\neq
m}\int_{D_{j}}\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x,z_{j}+t(y-z_{j}))\cdot(y-z_{j})\,dt\,v_{j}(y)\,dy$
$\displaystyle+$ $\displaystyle k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{j\neq
m}\int_{0}^{1}\underset{x}{\nabla}\Phi_{k}(z_{m}+t(x-z_{m}),z_{j})\cdot(x-z_{m})dt\,\int_{D_{j}}v_{j}(y)\,dy$
$\displaystyle-$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,b_{0}(z_{m})\,A^{0}\left(v_{m}\right)(x)-k^{2}\,\frac{1}{a_{0}}\,b_{0}(z_{m})\;\bm{E}\;\int_{D_{m}}v_{m}(y)\,dy$
$\displaystyle-$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,b_{0}(z_{m})\,\sum_{j\neq
m}\Phi_{k}(z_{m},z_{j})\int_{D_{j}}v_{j}(y)\,dy$

Let $w$ to be:

$w:=k^{2}\,\frac{1}{a_{0}}\,\epsilon_{p}\,\left[I-k^{2}\,\frac{1}{a_{0}}\,\epsilon_{p}\,A^{0}\right]^{-1}(1)=\left[\frac{a_{0}}{k^{2}\,b}I-A^{0}\right]^{-1}(1)$

and its corresponding scattering coefficient $C_{m}$ defined as:

$C_{m}:=\int_{D_{m}}w(x)\,dx.$

Then successively, for the equation $(\ref{equa1})$, dividing both sides by
$k^{2}\,\frac{b}{a_{0}}$, taking the inverse operator of
$\left[\displaystyle\frac{a_{0}}{k^{2}\,b}I-A^{0}\right]$ and integrating over
$D_{m}$, we obtain

$\displaystyle\left[1-C_{m}\;\bm{E}\right]\int_{D_{m}}v_{m}(y)\,dy$
$\displaystyle-$ $\displaystyle C_{m}\;\sum_{j\neq
m}\Phi_{k}(z_{m},z_{j})\,\int_{D_{j}}v_{j}(y)\,dy$ $\displaystyle=$
$\displaystyle\frac{C_{m}\,a_{0}}{k^{2}\,b}u^{i}(z_{m})+\frac{a_{0}}{k^{2}\,b}\int_{D_{m}}w(x)Err_{1}(x)\,dx.$

By multiplying the last equation by $k^{2}\;\frac{b}{a_{0}}\;C_{m}^{-1}$, we
end up with:

(3.6)
$k^{2}\,\frac{b}{a_{0}}\,\left[C^{-1}_{m}-\;\bm{E}\right]\int_{D_{m}}v_{m}(y)\,dy-k^{2}\,\frac{b}{a_{0}}\;\sum_{j\neq
m}\Phi_{k}(z_{m},z_{j})\,\int_{D_{j}}v_{j}(y)\,dy=u^{i}(z_{m})+Err_{2,m},$

where:

(3.7) $Err_{2,m}:=C_{m}^{-1}\,\int_{D_{m}}w(x)Err_{1}(x)\,dx.$

To estimate the term $Err_{2,m}$, we use the coming Lemma.

######  Lemma 3.2.

The total field $v(\cdot)$ admits the following a priori estimation:

(3.8)
$\|v\|_{\mathbb{L}^{2}(D)}\leq|\log(\delta)|^{h}\,\|u\|_{\mathbb{L}^{2}(D)}.$

For the scattering function and scattering coefficient, we have the following
relations:

(3.9)
$\|w\|_{\mathbb{L}^{2}(D)}\leq\delta^{-1}\,|\log(\delta)|^{h-1}\quad\text{and}\quad
C_{m}=|\log(\delta)|^{h-1}.$

###### Proof.

The a priori estimate $(\ref{prioriest})$ is already proved in Section 4 of
[17]. If in $(\ref{prioriest})$, we take $u(\cdot)=\tau$, recalling that
$\tau\sim\delta^{-2}\,\left|\log(\delta)\right|^{-1}$, we deduce the first
part of $(\ref{W2DC})$. The second part can be proved using Cauchy-Schwartz
inequality and the obtained estimation for $\|w\|_{\mathbb{L}^{2}(D)}$. ∎

First, by combining $(\ref{Err1})$ and $(\ref{Err2})$, we derive the following
expression for $Err_{2,m}$,

$\displaystyle Err_{2,m}$ $\displaystyle=$ $\displaystyle
C_{m}^{-1}\,\int_{D_{m}}w(x)\,\int_{0}^{1}\,\nabla
u^{i}(z_{m}+t(x-z_{m}))\cdot(x-z_{m})\,dt\,dx$ $\displaystyle+$ $\displaystyle
C_{m}^{-1}\,\tau\,\int_{D_{m}}w(x)\int_{D_{m}}|x-y|^{2}\;\log(|x-y|)\,v_{m}(y)\,dy\,dx$
$\displaystyle+$
$\displaystyle\frac{k^{2}\,\tau}{C_{m}\,a_{0}}\,\int_{D_{m}}w(x)\sum_{j\neq
m}\int_{D_{j}}\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x,z_{j}+t(y-z_{j}))\cdot(y-z_{j})\,dt\,v_{j}(y)\,dy\,dx$
$\displaystyle+$
$\displaystyle\frac{k^{2}\,\tau}{C_{m}\,a_{0}}\,\int_{D_{m}}w(x)\sum_{j\neq
m}\int_{0}^{1}\underset{x}{\nabla}\Phi_{k}(z_{m}+t(x-z_{m}),z_{j})\cdot(x-z_{m})dt\,\int_{D_{j}}v_{j}(y)\,dy\,dx$
$\displaystyle-$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,b_{0}(z_{m})\,C^{-1}_{m}\,\int_{D_{m}}w(x)\cdot
A^{0}(v_{m})(x)\,dx-k^{2}\,\frac{1}{a_{0}}\,b_{0}(z_{m})\;\bm{E}\;\int_{D_{m}}v_{m}(y)\,dy$
$\displaystyle-$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,b_{0}(z_{m})\,\sum_{j\neq
m}\Phi_{k}\left(z_{m},z_{j}\right)\,\int_{D_{j}}v_{j}(y)\,dy.$

Next, we split the expression of $Err_{2,m}$ into seven terms, and then we
estimate them separately to get an estimation of $Err_{2,m}$.

  1.   2. $\ast)$

Estimation of $T_{1}$:

$\displaystyle T_{1}$ $\displaystyle:=$ $\displaystyle
C_{m}^{-1}\,\int_{D_{m}}w(x)\,\int_{0}^{1}\,\nabla
u^{i}(z_{m}+t(x-z_{m}))\cdot(x-z_{m})\,dt\,dx$ (3.10)
$\displaystyle\left|T_{1}\right|$
$\displaystyle\overset{(\ref{W2DC})}{\lesssim}$
$\displaystyle\left|\log(\delta)\right|^{1-h}\,\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\left\|\int_{0}^{1}\,\nabla
u^{i}(z_{m}+t(\cdot-z_{m}))\cdot(\cdot-
z_{m})\,dt\right\|_{\mathbb{L}^{2}(D_{m})}$ $\displaystyle\lesssim$
$\displaystyle\left|\log(\delta)\right|^{1-h}\,\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\delta^{2}\overset{(\ref{W2DC})}{=}\mathcal{O}\left(\delta\right).$

  3.   4. $\ast)$

Estimation of $T_{2}$:

$\displaystyle T_{2}$ $\displaystyle:=$ $\displaystyle
C_{m}^{-1}\,\tau\,\int_{D_{m}}w(x)\int_{D_{m}}|x-y|^{2}\;\log(|x-y|)\,v_{m}(y)\,dy\,dx$
(3.11) $\displaystyle\left|T_{2}\right|$
$\displaystyle\overset{(\ref{W2DC})}{\lesssim}$
$\displaystyle\delta^{-3}\,\left|\log(\delta)\right|^{-1}\,\left\|\int_{D_{m}}|\cdot-y|^{2}\;\log(|\cdot-y|)\,v_{m}(y)\,dy\right\|_{\mathbb{L}^{2}(D_{m})}$
$\displaystyle\lesssim$
$\displaystyle\delta\,\left\|v_{m}\right\|_{\mathbb{L}^{2}(D_{m})}\overset{(\ref{prioriest})}{=}\mathcal{O}\left(\delta^{2}\,\left|\log(\delta)\right|^{h}\right).$

  5.   6. $\ast)$

Estimation of $T_{3}$:

$\displaystyle T_{3}$ $\displaystyle:=$
$\displaystyle\frac{k^{2}\,\tau}{C_{m}\,a_{0}}\,\int_{D_{m}}w(x)\sum_{j\neq
m}\int_{D_{j}}\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x,z_{j}+t(y-z_{j}))\cdot(y-z_{j})\,dt\,v_{j}(y)\,dy\,dx$
$\displaystyle\left|T_{3}\right|$ $\displaystyle\lesssim$
$\displaystyle\frac{\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}}{\delta\,\left|\log(\delta)\right|^{h}}\,\sum_{j\neq
m}\left[\int_{D_{m}}\int_{D_{j}}\int_{0}^{1}\left|\underset{y}{\nabla}\Phi_{k}(x,z_{j}+t(y-z_{j}))\right|^{2}dtdy\,dx\right]^{\frac{1}{2}}\,\left\|v_{j}\right\|_{\mathbb{L}^{2}(D_{j})}.$

Thanks to Lemma 3.1, we can approximate the previous estimation by:

$\displaystyle\left|T_{3}\right|$ $\displaystyle\lesssim$
$\displaystyle\frac{\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}}{\delta\,\left|\log(\delta)\right|^{h}}\,\sum_{j\neq
m}\left[\int_{D_{m}}\int_{D_{j}}\int_{0}^{1}\frac{1}{\left|x-(z_{j}+t(y-z_{j}))\right|^{2}}dtdy\,dx\right]^{\frac{1}{2}}\,\left\|v_{j}\right\|_{\mathbb{L}^{2}(D_{j})}$
$\displaystyle\lesssim$
$\displaystyle\frac{\delta\,\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}}{\left|\log(\delta)\right|^{h}}\,\sum_{j\neq
m}\frac{1}{\left|z_{m}-z_{j}\right|}\,\left\|v_{j}\right\|_{\mathbb{L}^{2}(D_{j})}$
$\displaystyle\lesssim$
$\displaystyle\frac{\delta\,\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}}{\left|\log(\delta)\right|^{h}}\,\left(\sum_{j\neq
m}\frac{1}{\left|z_{m}-z_{j}\right|^{2}}\right)^{\frac{1}{2}}\,\left\|v\right\|_{\mathbb{L}^{2}(D)}$
$\displaystyle\lesssim$
$\displaystyle\frac{\delta\,\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}}{\left|\log(\delta)\right|^{h}}\,d^{-1}\,\left\|v\right\|_{\mathbb{L}^{2}(D)}.$

Using the a priori estimate $(\ref{prioriest})$ and the estimation of
$\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}$, see $(\ref{W2DC})$, we get:

(3.12)
$\left|T_{3}\right|=\mathcal{O}\left(\left|\log(\delta)\right|^{h-1}\;\delta\;d^{-1}\right).$

  7.   8. $\ast)$

Estimation of $T_{4}$:

$T_{4}:=\frac{k^{2}\,\tau}{C_{m}\,a_{0}}\,\int_{D_{m}}w(x)\sum_{j\neq
m}\int_{0}^{1}\underset{x}{\nabla}\Phi_{k}(z_{m}+t(x-z_{m}),z_{j})\cdot(x-z_{m})dt\,\int_{D_{j}}v_{j}(y)\,dy\,dx.$

The above expression for $T_{4}$ is almost the same as that of $T_{3}$,
consequently its estimation can be handled in similar manner and we deduce,
from $(\ref{T42D})$, that:

$\left|T_{4}\right|=\mathcal{O}\left(\left|\log(\delta)\right|^{h-1}\;\delta\;d^{-1}\right).$

  9. $\ast)$

Estimation of $T_{5}$:

$\displaystyle T_{5}$ $\displaystyle:=$
$\displaystyle-k^{2}\,\frac{1}{a_{0}}\,b_{0}(z_{m})\,C^{-1}_{m}\,\int_{D_{m}}w(x)\cdot
A^{0}(v_{m})(x)\,dx$ $\displaystyle\left|T_{5}\right|$ $\displaystyle\lesssim$
$\displaystyle\left|\log(\delta)\right|^{1-h}\,\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\left\|A^{0}(v_{m})\right\|_{\mathbb{L}^{2}(D_{m})}$
$\displaystyle\lesssim$
$\displaystyle\left|\log(\delta)\right|^{2-h}\,\delta^{2}\,\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}\,\left\|v_{m}\right\|_{\mathbb{L}^{2}(D_{m})}.$

Then, thanks to Lemma 3.2, we deduce that:

$T_{5}=\mathcal{O}\left(\left|\log(\delta)\right|^{h+1}\,\delta^{2}\right).$

  10. $\ast)$

Estimation of $T_{6}$:

$\displaystyle T_{6}$ $\displaystyle:=$
$\displaystyle-k^{2}\,\frac{1}{a_{0}}\,b_{0}(z_{m})\;\bm{E}\;\int_{D_{m}}v_{m}(y)\,dy$
$\displaystyle\left|T_{6}\right|$ $\displaystyle\lesssim$
$\displaystyle\left\|1\right\|_{\mathbb{L}^{2}(D_{m})}\,\left\|v_{m}\right\|_{\mathbb{L}^{2}(D_{m})}\overset{(\ref{prioriest})}{=}\mathcal{O}\left(\delta^{2}\,\left|\log(\delta)\right|^{h}\right).$

  11. $\ast)$

Estimation of $T_{7}$:

$\displaystyle T_{7}$ $\displaystyle:=$
$\displaystyle-k^{2}\,\frac{1}{a_{0}}\,b_{0}(z_{m})\,\sum_{j\neq
m}\Phi_{k}\left(z_{m},z_{j}\right)\,\int_{D_{j}}v_{j}(y)\,dy$
$\displaystyle\left|T_{7}\right|$ $\displaystyle\lesssim$
$\displaystyle\sum_{j\neq
m}\log\left(\frac{1}{\left|z_{m}-z_{j}\right|}\right)\,\left\|1\right\|_{\mathbb{L}^{2}(D_{j})}\,\left\|v_{j}\right\|_{\mathbb{L}^{2}(D_{j})}$
$\displaystyle\overset{(\ref{prioriest})}{\lesssim}$
$\displaystyle\delta^{2}\,\left|\log(\delta)\right|^{h}\,\sum_{j\neq
m}\log\left(\frac{1}{\left|z_{m}-z_{j}\right|}\right).$

It is proven, in Lemma 2.6 of [17], that:

(3.13) $\sum_{j\neq m}\log\left(\frac{1}{d_{mj}}\right)=\log(1/d).$

Then,

(3.14)
$T_{7}=\mathcal{O}\left(\delta^{2}\,\left|\log(\delta)\right|^{h}\,\log(1/d)\right).$

Finally, by gathering $(\ref{T12D})-(\ref{T52D})$, we end up with the
following estimation for $Err_{2,m}$.

$Err_{2,m}=\sum_{i=1}^{7}T_{i}=\mathcal{O}\left(\delta\,\left|\log(\delta)\right|^{h-1}\,d^{-1}\right),$

and, knowing that $d\sim e^{-\,\left|\log(\delta)\right|^{t}}$, we obtain:

(3.15)
$Err_{2,m}=\mathcal{O}\left(\delta\,\left|\log(\delta)\right|^{h-1}\,e^{\left|\log(\delta)\right|^{t}}\right).$

At this stage, the algebraic system given by $(\ref{AS2D})$ can be written as:

$k^{2}\,\frac{b}{a_{0}}\,\left[C^{-1}_{m}-\bm{E}\right]\int_{D_{m}}v_{m}(y)\,dy-k^{2}\,\frac{b}{a_{0}}\;\sum_{j\neq
m}\Phi_{k}(z_{m},z_{j})\int_{D_{j}}v_{j}(y)\,dy=u^{i}(z_{m})+Err_{2,m}.$

We set:

$Q_{m}:=k^{2}\,\frac{b}{a_{0}}\,\left[C^{-1}_{m}-\bm{E}\right]\int_{D_{m}}v_{m}(y)\,dy.$

Then

(3.16) $Q_{m}-\sum_{j\neq
m}\Phi_{k}(z_{m},z_{j})\,\left[C^{-1}_{j}-\bm{E}\right]^{-1}Q_{j}=u^{i}(z_{m})+Err_{2,m}.$

To write short, we set

(3.17) $C_{m}^{\star}:=\left[C^{-1}_{m}-\bm{E}\right]^{-1},\;\;m=1,\cdots,M,$

hence, the equation $(\ref{slcc})$ will takes the following form,

(3.18) $Q_{m}-\sum_{j\neq
m}\Phi_{k}(z_{m},z_{j})\;C_{j}^{\star}\;Q_{j}=u^{i}(z_{m})+Err_{2,m},$

which can be rewritten in a in matrix form as

(3.19) $\left(I-B_{k}\right)\cdot Q=U+Err,$

where $B_{k}=\left(B_{k,mj}\right)_{m,j=1}^{M}$ such that

(3.20)
$B_{k,mj}:=\Phi_{k}(z_{m};z_{j})\;C_{j}^{\star}\;\left(1-\underset{j,m}{\bm{\delta}}\right),$

$U=\left(u^{i}(z_{j})\right)_{j=1,\cdots,M}$ and
$Err=\left(Err_{2,m},\cdots,Err_{2,m}\right)$.

As it was done in the proof of Theorem 1.1, we start by looking under what
conditions the following non perturbed linear system

(3.21) $\left(I-B_{k}\right)\cdot\tilde{Q}=U$

is invertible. To accomplish this, let us evaluate the norm of $B_{k}$. For
this we have:

$\left\|B_{k}\right\|=\max_{m}\,\sum_{j\neq
m}\left|B_{k,mj}\right|\overset{(\ref{defB})}{=}\max_{m}\,\sum_{j\neq
m}\left|\Phi_{k}(z_{m},z_{j})\;C_{j}^{\star}\right|\overset{(\ref{defCstar})}{=}\max_{m}\,\sum_{j\neq
m}\frac{\left|\Phi_{k}(z_{m},z_{j})\right|}{\left|C^{-1}_{j}-\bm{E}\right|}.$

Thanks to $(\ref{W2DC})$, we know that
$C_{j}\sim\left|\log(\delta)\right|^{h-1}$ and from Lemma 3.1 we can
approximate $\Phi_{k}(\cdot,\cdot)$ by the logarithmic kernel. We manage all
this to deduce:

(3.22) $\left\|B_{k}\right\|\lesssim|\log(\delta)|^{h-1}\,\sum_{j\neq
m}\log\left(\frac{1}{d_{mj}}\right)\overset{(\ref{sumlog})}{=}\mathcal{O}\left(|\log(\delta)|^{h-1}\,\log\left(\frac{1}{d}\right)\right),$

and, the condition $\|B_{k}\|<1$ will be fulfilled if:

$\log(1/d)<|\log(\delta)|^{1-h}.$

This implies that:

(3.23) $d>\exp\left(-|\log(\delta)|^{1-h}\right).$

Now, we assume that the invertibility condition $(\ref{Condinv})$ is
fulfilled, by subtracting $(\ref{nonper})$ from $(\ref{per})$, we get

$Q-\tilde{Q}=\left(I-B_{k}\right)^{-1}\,Err$

and, by taking the Euclidean norm in both sides, we obtain

(3.24) $\|Q-\tilde{Q}\|\lesssim\|Err\|=\mathcal{O}\left(Err_{2,m}\right).$

To prove $(\ref{intro-usca-2D})$ we derive an approximation for the scattered
field. To do this, we start by setting the coming L.S.E:

$u^{s}(x,\theta)=k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{m=1}^{M}\,\int_{D_{m}}\Phi_{k}(x,y)\;v_{m}(y)\;dy,$

which, after using Taylor expansion near the center $z_{m}$ for the kernel
$\Phi_{k}(\cdot,\cdot)$, becomes:

$\displaystyle u^{s}(x,\theta)$ $\displaystyle=$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{m=1}^{M}\;\Phi_{k}(x,z_{m})\;\int_{D_{m}}v_{m}(y)\,dy$
$\displaystyle+$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{m=1}^{M}\,\int_{D_{m}}\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x,z_{m}+t(y-z_{m}))\cdot(y-z_{m})\,dt\;v_{m}(y)\,dy.$

We estimate the second term on the right hand side as:

$\displaystyle J_{1}$ $\displaystyle:=$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{m=1}^{M}\,\int_{D_{m}}\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x,z_{m}+t(y-z_{m}))\cdot(y-z_{m})\,dt\;v_{m}(y)\,dy$
$\displaystyle\left|J_{1}\right|$ $\displaystyle\lesssim$
$\displaystyle\tau\sum_{m=1}^{M}\left\|\int_{0}^{1}\underset{y}{\nabla}\Phi_{k}(x,z_{m}+t(\cdot-
z_{m}))\cdot(\cdot-
z_{m})dt\right\|_{\mathbb{L}^{2}(D_{m})}\left\|v_{m}\right\|_{\mathbb{L}^{2}(D_{m})}.$

We recall that $\tau\sim\delta^{-2}\left|\log(\delta)\right|^{-1}$ and we
assume that the point $x$ is away from $D$, this implies the smoothness of the
function under the integral sign. Then,

$\left|J_{1}\right|\lesssim\left|\log(\delta)\right|^{-1}\,\sum_{m=1}^{M}\left\|v_{m}\right\|_{\mathbb{L}^{2}(D_{m})}\leq\left|\log(\delta)\right|^{-1}\,\left\|v\right\|_{\mathbb{L}^{2}(D)}\overset{(\ref{prioriest})}{=}\mathcal{O}\left(\delta\,\left|\log(\delta)\right|^{h-1}\right).$

Therefore,

(3.25) $\displaystyle u^{s}(x,\theta)$ $\displaystyle=$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\tau\,\sum_{m=1}^{M}\,\Phi_{k}(x,z_{m})\,\int_{D_{m}}v_{m}(y)\,dy+\mathcal{O}\left(\delta\,\left|\log(\delta)\right|^{h-1}\right)$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{M}\,\Phi_{k}(x,z_{m})\,C_{m}^{\star}\,Q_{m}+\mathcal{O}\left(\delta\,\left|\log(\delta)\right|^{h-1}\right)$
$\displaystyle=$
$\displaystyle\langle\,\Phi_{k}(x,z)\,\,C^{\star};\tilde{Q}\rangle+\langle\,\Phi_{k}(x,z)\,\,C^{\star};\,(Q-\tilde{Q})\rangle+\mathcal{O}\left(\delta\,\left|\log(\delta)\right|^{h-1}\right).$

We need to estimate the second term on the right hand side of the previous
equation. To do this, we set
$J_{2}:=\langle\,\Phi_{k}(x,z)\,\,C^{\star}\,;\,(Q-\tilde{Q})\rangle$. Then,

$|J_{2}|\leq\|\,\Phi_{k}(x,z)\,C^{\star}\,\|\,\|Q-\tilde{Q}\|\overset{(\ref{DiffQ-Q})}{\leq}\|\,\Phi_{k}(x,z)\,\,C^{\star}\,\|\;\delta^{1-t}\;|\log(\delta)|^{h-1}.$

Again, we suppose that the point $x$ is away from the domain $D$, which
implies the smoothness of the functions $\,\Phi_{k}(x,z_{m})$, $m=1;\cdots;M$,
then

$\left|J_{2}\right|\lesssim\left[\sum_{m=1}^{M}\left|C_{m}^{\star}\,\right|^{2}\right]^{\frac{1}{2}}\;\delta^{1-t}\;|\log(\delta)|^{h-1}\lesssim\,|\log(\delta)|^{h-1}\;\delta^{1-t}\;|\log(\delta)|^{h-1}.$

Then,

(3.26) $J_{2}=\mathcal{O}\left(\delta^{1-t}\;|\log(\delta)|^{2(h-1)}\right).$

Finally, $(\ref{SK})$ becomes,

(3.27)
$u^{s}(x,\theta)=\langle\,\Phi_{k}(x,z)\,\,C^{\star}\,;\tilde{Q}\rangle+\mathcal{O}\left(\delta^{1-t}\,|\log(\delta)|^{2\,(h-1)}\right)+\mathcal{O}\left(\delta\,|\log(\delta)|^{(h-1)}\right).$

This ends the proof of Theorem 1.4.
Now, we define the truncated series that we denote by
$\tilde{Q}^{N},N\in\mathbb{N},$ by
$\tilde{Q}^{N}:=\underset{n=0}{\overset{N}{\sum}}B_{k}^{n}\cdot U$ and its
associated scattered field

(3.28)
$u^{s,N}(x,\theta):=\langle\,\Phi_{k}(x,z)\,C^{\star}\,;\tilde{Q}^{N}\rangle.$

Then, by subtracting $(\ref{31M})$ from $(\ref{T+R1+R2})$, we obtain

$\displaystyle u^{s}(x,\theta)-u^{s,N}(x,\theta)$ $\displaystyle=$
$\displaystyle\langle\,\Phi_{k}(x,z)\,\,C^{\star}\,;\left(\tilde{Q}-\tilde{Q}^{N}\right)\rangle$
$\displaystyle+$
$\displaystyle\mathcal{O}\left(\delta^{1-t}\,|\log(\delta)|^{2\,(h-1)}\right)+\mathcal{O}\left(\delta\,|\log(\delta)|^{(h-1)}\right)$
$\displaystyle=$
$\displaystyle\langle\,\Phi_{k}(x,z)\,\,C^{\star}\,;\left(\sum_{n\geq
N+1}B_{k}^{n}\right)\cdot U\rangle$ $\displaystyle+$
$\displaystyle\mathcal{O}\left(\delta^{1-t}\,|\log(\delta)|^{2\,(h-1)}\right)+\mathcal{O}\left(\delta\,|\log(\delta)|^{(h-1)}\right).$

After taking the modulus, in both sides of the previous equation, we obtain

$\displaystyle\left|u^{s}(x,\theta)-u^{s,N}(x,\theta)\right|$
$\displaystyle\leq$
$\displaystyle\left|\,\Phi_{k}(x,z)\,\right|\,\left|C^{\star}\right|\;\sum_{n\geq
N+1}\left|B_{k}\right|^{n}\,\left|U\right|$ $\displaystyle+$
$\displaystyle\mathcal{O}\left(\delta^{1-t}\,|\log(\delta)|^{2\,(h-1)}\right)+\mathcal{O}\left(\delta\,|\log(\delta)|^{(h-1)}\right).$

Because, $\left|U\right|\sim 1$ and $C^{\star}\sim
C\sim\left|\log(\delta)\right|^{h-1}$, we reduce the previous inequality to

$\displaystyle\left|u^{s}(x,\theta)-u^{s,N}(x,\theta)\right|$
$\displaystyle\lesssim$
$\displaystyle\left|\log(\delta)\right|^{h-1}\,\underset{z_{j}}{\min}\left|\Phi_{k}(x,z_{j})\right|\;\sum_{n\geq
N+1}\left|B_{k}\right|^{n}$ $\displaystyle+$
$\displaystyle\mathcal{O}\left(\delta^{1-t}\,|\log(\delta)|^{2\,(h-1)}\right)+\mathcal{O}\left(\delta\,|\log(\delta)|^{(h-1)}\right).$

As $\underset{z_{j}}{\min}\left|\Phi_{k}(x,z_{j})\right|\sim 1$, we obtain

$\displaystyle\left|u^{s}(x,\theta)-u^{s,N}(x,\theta)\right|$
$\displaystyle\lesssim$
$\displaystyle\left|\log(\delta)\right|^{h-1}\,\sum_{n\geq
N+1}\left|B_{k}\right|^{n}$ $\displaystyle+$
$\displaystyle\mathcal{O}\left(\delta^{1-t}\,|\log(\delta)|^{2\,(h-1)}\right)+\mathcal{O}\left(\delta\,|\log(\delta)|^{(h-1)}\right).$

From the expression of the matrix $B_{k}$, see $(\ref{defB})$, we can prove
that
$\left|B_{k}\right|=\mathcal{O}\left(\left|\log(\delta)\right|^{h-1}\,\left|\log\left(\frac{1}{d}\right)\right|\right)$
and using the fact that $d\sim e^{-\left|\log(\delta)\right|^{t}}$, we deduce
that
$\left|B_{k}\right|=\mathcal{O}\left(\left|\log(\delta)\right|^{t+h-1}\right)$.
Hence,

$\displaystyle\left|u^{s}(x,\theta)-u^{s,N}(x,\theta)\right|$
$\displaystyle\lesssim$
$\displaystyle\left|\log(\delta)\right|^{h-1}\,\sum_{n\geq
N+1}\left|\log(\delta)\right|^{n(t+h-1)}$ $\displaystyle+$
$\displaystyle\mathcal{O}\left(\delta^{1-t}\,|\log(\delta)|^{2\,(h-1)}\right)+\mathcal{O}\left(\delta\,|\log(\delta)|^{(h-1)}\right).$

Finally,

  1. $\ast)$

under the condition $1-t-h>0$,

$\left|u^{s}(x,\theta)-u^{s,N}(x,\theta)\right|=\mathcal{O}\left(\left|\log(\delta)\right|^{(h-1)+(N+1)(t+h-1)}\right).$

This proves $(\ref{intro-N-interactions-proof-2D})$.

  2. $\ast)$

under the condition $1-t-h=0$, the equation $(\ref{T+R1+R2})$ becomes

$u^{s}(x,\theta)=\langle\,\Phi_{k}(x,z)\,\,C^{\star}\,;\tilde{Q}\rangle+\mathcal{O}\left(\delta^{1-t}\,|\log(\delta)|^{-2\,t}\right),$

and we set
$u^{s,\infty}(x,\theta):=\langle\,\Phi_{k}(x,z)\,\,C^{\star}\,;\tilde{Q}\rangle$,
then:

$u^{s}(x,\theta)-u^{s,\infty}(x,\theta)=\mathcal{O}\left(\delta^{1-t}\,|\log(\delta)|^{-2\,t}\right),$

this proves $(\ref{intro-infty-interactions-proof-2D})$. Form the definition
of $u^{s,\infty}(\cdot,\cdot)$ we have

$\displaystyle u^{s,\infty}(x,\theta)$ $\displaystyle=$
$\displaystyle\langle\,\Phi_{k}(x,z)\,\,C^{\star}\,;\tilde{Q}\rangle$
$\displaystyle=$ $\displaystyle\sum_{n\geq
0}\langle\,\Phi_{k}(x,z)\,\,C^{\star}\,;B_{k}^{n}\cdot U\rangle$
$\displaystyle\sim$
$\displaystyle\left|\log(\delta)\right|^{h-1}\sim\left|\log(\delta)\right|^{-t}>>\delta^{1-t}\,|\log(\delta)|^{-2\,t}.$

In addition,

$\displaystyle u^{s,N}(x,\theta)-u^{s,N-1}(x,\theta)$ $\displaystyle=$
$\displaystyle\langle\Phi_{k}(x,z)C^{\star};\left(\tilde{Q}^{N}-\tilde{Q}^{N-1}\right)\rangle$
$\displaystyle\left|u^{s,N}(x,\theta)-u^{s,N-1}(x,\theta)\right|$
$\displaystyle\lesssim$
$\displaystyle\left|C^{\star}\right|\;\left|\tilde{Q}^{N}-\tilde{Q}^{N-1}\right|\lesssim\left|C\right|\;\left|B_{k}\right|^{N}.$

Using the estimation of $C\sim\left|\log(\delta)\right|^{h-1}$ and
$\left|B_{k}\right|=\mathcal{O}\left(\left|\log(\delta)\right|^{t+h-1}\right)$
we obtain:

$\left|u^{s,N}(x,\theta)-u^{s,N-1}(x,\theta)\right|=\mathcal{O}\left(\left|\log(\delta)\right|^{(h-1)-N(1-h-t)}\right).$

This concludes the proof of Corollary 1.5.

##  4\. Appendix

###  4.1. Estimation of the scattering coefficient $C_{m}$ and
$\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}$

We recall that:

$C_{m}:=\int_{D_{m}}w(x)\,dx,$

where $w$ is solution, in the domain $D$, of

(4.1)
$\left[A^{0}-\left(\lambda_{n_{0}}+\mathcal{O}\left(\delta^{2+h}\right)\right)\right]w=1.$

We can write $C_{m}$, by spectral decomposition of $w$, as:

(4.2) $C_{m}:=\int_{D_{m}}w(x)\,dx=\sum_{n=1}^{+\infty}\langle
w;e_{n}\rangle_{\mathbb{L}^{2}(D_{m})}\;\int_{D_{m}}e_{n}(x)\;dx.$

In addition, we have

$\displaystyle\int_{D_{m}}e_{n}(x)\,dx=\int_{D_{m}}1\;e_{n}(x)\,dx$
$\displaystyle\overset{(\ref{EquaW})}{=}$
$\displaystyle\int_{D_{m}}\left[A^{0}-\left(\lambda_{n_{0}}+\mathcal{O}\left(\delta^{2+h}\right)\right)\right](w)(x)\;e_{n}(x)\;dx$
$\displaystyle=$
$\displaystyle\langle\left[A^{0}-\left(\lambda_{n_{0}}+\mathcal{O}\left(\delta^{2+h}\right)\right)\right]w;e_{n}\rangle_{\mathbb{L}^{2}(D_{m})}$
$\displaystyle=$ $\displaystyle\langle
A^{0}w;e_{n}\rangle_{\mathbb{L}^{2}(D_{m})}-\left(\lambda_{n_{0}}+\mathcal{O}\left(\delta^{2+h}\right)\right)\,\langle
w;e_{n}\rangle_{\mathbb{L}^{2}(D_{m})}.$

Using the fact that $A^{0}$ is a self adjoint operator with
$\left(\lambda_{n};e_{n}\right)$ as an eigen-system we derive, from the
previous equation, the coming relation linking the Fourier coefficient of $w$
with the Fourier coefficient of function unity $1$:

(4.3) $\langle w;e_{n}\rangle_{\mathbb{L}^{2}(D_{m})}=\frac{\langle
1;e_{n}\rangle_{\mathbb{L}^{2}(D_{m})}}{\left(\lambda_{n}-\left(\lambda_{n_{0}}+\mathcal{O}\left(\delta^{2+h}\right)\right)\right)}.$

Then, by combining $(\ref{wserie})$ with $(\ref{coeffw})$, we obtain

(4.4) $C_{m}=\sum_{n=1}^{+\infty}\frac{\left(\langle
1;e_{n}\rangle_{\mathbb{L}^{2}(D_{m})}\right)^{2}}{\left(\lambda_{n}-\left(\lambda_{n_{0}}+\mathcal{O}\left(\delta^{2+h}\right)\right)\right)}.$

To see the behaviour of $\langle 1;e_{n}\rangle_{\mathbb{L}^{2}(D_{m})}$, with
respect to the radius of $D_{m}$, i.e. $\delta$, we start by scaling:

$\int_{D_{m}}e_{n}(x)\;dx=\delta^{3}\;\int_{B}e_{n}(z_{m}+\delta\,\xi)\;d\xi=\delta^{3}\;\int_{B_{m}}\tilde{e}_{n}(\xi)\;d\xi.$

We define the following orthonormal basis:

$\overline{e}_{n}=\frac{\tilde{e}_{n}}{\|\tilde{e}_{n}\|_{\mathbb{L}^{2}(B)}},$

where, in effortless manner, we can prove that
$\|\tilde{e}_{n}\|_{\mathbb{L}^{2}(B)}=\delta^{-\frac{3}{2}}$. Synthesizing
what precedes we deduce:

(4.5)
$\int_{D_{m}}e_{n}(x)\;dx=\delta^{\frac{3}{2}}\;\int_{B}\overline{e}_{n}(\xi)\;d\xi.$

Using the derived scales, the equation $(\ref{RMSeries})$ becomes

$C_{m}=\delta^{3}\sum_{n\geq 1}\frac{\left(\langle
1;\overline{e}_{n}\rangle_{\mathbb{L}^{2}(B)}\right)^{2}}{\lambda_{n}-(\lambda_{n_{0}}+\mathcal{O}(\delta^{2+h}))},$

and, by scaling the eigenvalues $\lambda_{n}=\delta^{2}\,\tilde{\lambda}_{n}$
and $\lambda_{n_{0}}=\delta^{2}\,\tilde{\lambda}_{n_{0}}$, we obtain

(4.6) $C_{m}=\delta\,\sum_{n\geq 1}\frac{\left(\langle
1;\overline{e}_{n}\rangle_{\mathbb{L}^{2}(B)}\right)^{2}}{\tilde{\lambda}_{n}-(\tilde{\lambda}_{n_{0}}+\mathcal{O}(\delta^{h}))}=\delta^{1-h}\left(\langle
1;\overline{e}_{n_{0}}\rangle_{\mathbb{L}^{2}(B)}\right)^{2}+\delta\sum_{n\neq
n_{0}}\frac{\left(\langle
1;\overline{e}_{n}\rangle_{\mathbb{L}^{2}(B)}\right)^{2}}{\tilde{\lambda}_{n}-\left(\tilde{\lambda}_{n_{0}}+\mathcal{O}\left(\delta^{h}\right)\right)}.$

The preceding formula is consistent if

$\sum_{n}\left(\langle
1;\overline{e}_{n}\rangle_{\mathbb{L}^{2}(B)}\right)^{2}\;<\infty.$

Now, we need to show the convergence of the previous series. For this, let
$\beta$ such that

$\beta-\underset{n}{\max}\,\,\lambda_{n}>>1$

and $w_{\beta}$ solution of

(4.7) $\left(A^{0}-\beta\right)\,w_{\beta}=1.$

We have:

$\left(\langle
1;\overline{e}_{n}\rangle_{\mathbb{L}^{2}(B)}\right)^{2}=\left(\beta-\lambda_{n}\right)\;\frac{\left(\langle
1;\overline{e}_{n}\rangle_{\mathbb{L}^{2}(B)}\right)^{2}}{\left(\beta-\lambda_{n}\right)}\leq\beta\;\frac{\left(\langle
1;\overline{e}_{n}\rangle_{\mathbb{L}^{2}(B)}\right)^{2}}{\left(\beta-\lambda_{n}\right)}.$

Hence, after taking the series with respect to $n$ on both sides, we get

$\sum_{n}\left(\langle
1;\overline{e}_{n}\rangle_{\mathbb{L}^{2}(B)}\right)^{2}\,\leq\,\beta\;\sum_{n}\frac{\left(\langle
1;\overline{e}_{n}\rangle_{\mathbb{L}^{2}(B)}\right)^{2}}{\left(\beta-\lambda_{n}\right)}.$

In addition, using $(\ref{wbeta})$, we can derive that

$\sum_{n}\frac{\left(\langle
1;\overline{e}_{n}\rangle_{\mathbb{L}^{2}(B)}\right)^{2}}{\left(\beta-\lambda_{n}\right)}=-\int_{D}w_{\beta}(x)\;dx<\infty.$

Which proves that $(\ref{intw})$ is well defined. Finally, we deduce that

(4.8) $C_{m}:=\int_{D_{m}}w(x)\,dx=\mathcal{O}\left(\delta^{1-h}\right).$

To estimate the $\mathbb{L}^{2}-$norm of $w$, we start by writing its series
expansion given by

$\left\|w\right\|^{2}_{\mathbb{L}^{2}(D_{m})}=\sum_{n=1}^{+\infty}\left|\langle
w;e_{n}\rangle_{\mathbb{L}^{2}(D_{m})}\right|^{2}\overset{(\ref{coeffw})}{=}\sum_{n}\frac{\left|\langle
1;e_{n}\rangle_{\mathbb{L}^{2}(D_{m})}\right|^{2}}{\left|\left(\lambda_{n}-\left(\lambda_{n_{0}}+\mathcal{O}\left(\delta^{2+h}\right)\right)\right)\right|^{2}}.$

We use the scale of the eigenvalues $\lambda_{n}$, which is of order
$\delta^{2}$, and the scale of the eigenfunction integral, given by
$(\ref{scaleeigfct})$, to obtain:

$\left\|w\right\|^{2}_{\mathbb{L}^{2}(D_{m})}=\delta^{-1}\;\sum_{n}\frac{\left|\langle
1;\overline{e}_{n}\rangle_{\mathbb{L}^{2}(B)}\right|^{2}}{\left|\left(\tilde{\lambda}_{n}-\left(\tilde{\lambda}_{n_{0}}+\mathcal{O}\left(\delta^{h}\right)\right)\right)\right|^{2}}=\mathcal{O}\left(\delta^{-1-2h}\right).$

Finally, we deduce that:

(4.9)
$\left\|w\right\|_{\mathbb{L}^{2}(D_{m})}=\mathcal{O}\left(\delta^{-\frac{1}{2}-h}\right).$

###  4.2. Proof of Proposition 2.1

We start by setting the L.S.E, in presence of several particles, given by

$\displaystyle v_{i}(x)$ $\displaystyle-$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\int_{D_{i}}\Phi_{k}(x,y)\,(b-b_{0}(y))\,v_{i}(y)\,dy$
$\displaystyle-$ $\displaystyle k^{2}\,\frac{1}{a_{0}}\,\sum_{m\neq
i}\int_{D_{m}}\Phi_{k}(x,y)\,(b-b_{0}(y))\,v_{m}(y)\,dy=u(x).$

By expanding the function $b_{0}(\cdot)$, near the centers
$z_{i},i=1,\cdots,M$, and rearranging the equation we obtain

$\displaystyle v_{i}(x)$ $\displaystyle-$ $\displaystyle
k^{2}\,\mu_{0}\,(b-b_{0}(z_{i}))\,\int_{D_{i}}\Phi_{0}(x,y)\,v_{i}(y)\,dy=u(x)$
$\displaystyle-$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\int_{D_{i}}\Phi_{0}(x,y)\int_{0}^{1}\nabla
b_{0}(z_{i}+t(y-z_{i}))\cdot(y-z_{i})\,dt\,v_{i}(y)\,dy$ $\displaystyle+$
$\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\int_{D_{i}}\left(\Phi_{k}-\Phi_{0}\right)(x,y)\,(b-b_{0}(y))\,v_{i}(y)\,dy$
$\displaystyle+$ $\displaystyle k^{2}\,\frac{1}{a_{0}}\,\sum_{m\neq
i}\int_{D_{m}}\Phi_{k}(x,y)\,(b-b_{0}(y))\,v_{m}(y)\,dy.$

After scaling the last equation, we get

$\displaystyle\left(I\,-\,k^{2}\,\mu_{0}\,(b-b_{0}(z_{i}))\,\delta^{2}\,A^{0}\right)\tilde{v}_{i}(\eta)=\tilde{u}_{i}(\eta)$
$\displaystyle-$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\delta^{3}\,\int_{B}\Phi_{0}(\eta,\xi)\,\int_{0}^{1}\widetilde{\nabla
b}_{0}\left(z_{i}+t\,\delta\,\xi\right)\cdot\xi\,dt\,\tilde{v}_{i}(\xi)\,d\xi$
$\displaystyle+$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\delta^{3}\,\int_{B}\left(\tilde{\Phi}_{k}-\tilde{\Phi}_{0}\right)(\eta,\xi)\,(b-\tilde{b_{0}}(\xi))\,\tilde{v}_{i}(\xi)\,d\xi$
$\displaystyle+$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\delta^{3}\,\sum_{m\neq
i}\int_{B}\tilde{\Phi}_{k}(\eta,\xi)\,(b-\tilde{b_{0}}(\xi))\,\tilde{v}_{m}(\xi)\,d\xi,$

where we recall that

$A^{0}(f)(x):=\int_{B}\Phi_{0}(x,y)\,f(y)\,dy.$

Then

$\displaystyle\tilde{v}_{i}$ $\displaystyle=$
$\displaystyle\lambda_{n_{0}}\,\mathcal{R}\left(\lambda_{n_{0}}+\delta^{h};A^{0}\right)\Bigg{[}\tilde{u}_{i}(\cdot)$
$\displaystyle-$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\delta^{3}\,\int_{B}\Phi_{0}(\cdot,\xi)\,\int_{0}^{1}\widetilde{\nabla
b}_{0}\left(z_{i}+t\,\delta\,\xi\right)\cdot\xi\,dt\,\tilde{v}_{i}(\xi)\,d\xi$
$\displaystyle+$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\delta^{3}\,\int_{B}\left(\tilde{\Phi}_{k}-\tilde{\Phi}_{0}\right)(\cdot,\xi)\,(b-\tilde{b_{0}}(\xi))\,\tilde{v}_{i}(\xi)\,d\xi$
$\displaystyle+$ $\displaystyle
k^{2}\,\frac{1}{a_{0}}\,\delta^{3}\,\sum_{m\neq
i}\int_{B}\tilde{\Phi}_{k}(\cdot,\xi)\,(b-\tilde{b_{0}}(\xi))\,\tilde{v}_{m}(\xi)\,d\xi\Bigg{]}.$

Now, by taking the $\|\cdot\|_{\mathbb{L}^{2}(B)}$ in both sides of the last
equality, we deduce

$\displaystyle\|\tilde{v}_{i}\|_{\mathbb{L}^{2}(B)}$ $\displaystyle\leq$
$\displaystyle\delta^{-h}\|\tilde{u}_{i}\|_{\mathbb{L}^{2}(B)}$
$\displaystyle+$
$\displaystyle\delta^{3}\;\delta^{-h}\left\|\int_{B}(\tilde{\Phi}_{k}-\tilde{\Phi}_{0})(\cdot,\xi)\,(b-\tilde{b_{0}}(\xi))\,\tilde{v}_{i}(\xi)\,d\xi\right\|_{\mathbb{L}^{2}(B)}$
$\displaystyle+$ $\displaystyle\delta^{3}\;\delta^{-h}\;\sum_{m\neq
i}\left\|\int_{B}\tilde{\Phi}_{k}(\cdot,\xi)\,(b-\tilde{b_{0}}(\xi))\,\tilde{v}_{m}(\xi)\,d\xi\right\|_{\mathbb{L}^{2}(B)}$
$\displaystyle+$
$\displaystyle\delta^{3}\;\delta^{-h}\left\|\int_{B}\Phi_{0}(\cdot,\xi)\int_{0}^{1}\widetilde{\nabla
b}_{0}(z_{i}+t\,\delta\,\xi)\cdot\xi\,dt\,\tilde{v}_{i}(\xi)\,d\xi\right\|_{\mathbb{L}^{2}(B)},$

hence,

$\left\|\tilde{v}_{i}\right\|_{\mathbb{L}^{2}(B)}\leq\delta^{-h}\,\left\|\tilde{u}_{i}\right\|_{\mathbb{L}^{2}(B)}+\delta^{1-h}\,\left\|\tilde{v}_{i}\right\|_{\mathbb{L}^{2}(B)}+\delta^{1-h}\,\sum_{m\neq
i}\left\|\tilde{v}_{m}\right\|_{\mathbb{L}^{2}(B)}+\delta^{3-h}\,\left\|\tilde{v}_{i}\right\|_{\mathbb{L}^{2}(B)},$

which can be reduced to

$\displaystyle\left\|\tilde{v}_{i}\right\|_{\mathbb{L}^{2}(B)}$
$\displaystyle\leq$
$\displaystyle\delta^{-h}\,\left\|\tilde{u}_{i}\right\|_{\mathbb{L}^{2}(B)}+\delta^{1-h}\,\left\|\tilde{v}_{i}\right\|_{\mathbb{L}^{2}(B)}+\delta^{1-h}\,\sum_{m\neq
i}\left\|\tilde{v}_{m}\right\|_{\mathbb{L}^{2}(B)}$
$\displaystyle\left\|\tilde{v}_{i}\right\|_{\mathbb{L}^{2}(B)}$
$\displaystyle\leq$
$\displaystyle\left(1-\delta^{1-h}\right)^{-1}\;\delta^{-h}\,\left\|\tilde{u}_{i}\right\|_{\mathbb{L}^{2}(B)}+\left(1-\delta^{1-h}\right)^{-1}\,\delta^{1-h}\;\sum_{m\neq
i}\left\|\tilde{v}_{m}\right\|_{\mathbb{L}^{2}(B)}$ $\displaystyle\lesssim$
$\displaystyle\delta^{-h}\,\left\|\tilde{u}_{i}\right\|_{\mathbb{L}^{2}(B)}+\delta^{1-h}\;\sum_{m\neq
i}\left\|\tilde{v}_{m}\right\|_{\mathbb{L}^{2}(B)}.$

Scaling back we obtain

$\left\|v_{i}\right\|_{\mathbb{L}^{2}(D_{i})}\leq\delta^{-h}\left\|u_{i}\right\|_{\mathbb{L}^{2}(D_{i})}+\delta^{1-h}\;\sum_{m\neq
i}\left\|v_{m}\right\|_{\mathbb{L}^{2}(D_{m})},$

and

$\left\|v_{i}\right\|_{\mathbb{L}^{2}(D_{i})}\leq\delta^{-h}\left\|u_{i}\right\|_{\mathbb{L}^{2}(D_{i})}+\delta^{1-h}\;\sum_{m=1}^{M}\left\|v_{m}\right\|_{\mathbb{L}^{2}(D_{m})}.$

By taking the modulus squared in both sides we obtain

$\left\|v_{i}\right\|^{2}_{\mathbb{L}^{2}(D_{i})}\lesssim\delta^{-2h}\,\left\|u_{i}\right\|^{2}_{\mathbb{L}^{2}(D_{i})}+\delta^{2(1-h)}\;\sum_{m=1}^{M}\left\|v_{m}\right\|^{2}_{\mathbb{L}^{2}(D_{m})}.$

Summing up with respect to the index $i$, we get

(4.10)
$\sum_{i=1}^{M}\,\left\|v_{i}\right\|^{2}_{\mathbb{L}^{2}(D_{i})}\lesssim\delta^{-2h}\,\sum_{i=1}^{M}\,\left\|u_{i}\right\|^{2}_{\mathbb{L}^{2}(D_{i})}+\delta^{2(1-h)}\;\sum_{m=1}^{M}\left\|v_{m}\right\|^{2}_{\mathbb{L}^{2}(D_{m})}.$

We deduce, under the condition $h<1$, that:

$\displaystyle\sum_{i=1}^{M}\,\left\|v_{i}\right\|^{2}_{\mathbb{L}^{2}(D_{i})}$
$\displaystyle\leq$
$\displaystyle\left(1-\delta^{2(1-h)}\right)^{-1}\;\;\delta^{-2h}\,\sum_{i=1}^{M}\,\left\|u_{i}\right\|^{2}_{\mathbb{L}^{2}(D_{i})}.$

This allows us, under the condition $h<1$, to deduce that

(4.11)
$\left\|v\right\|_{\mathbb{L}^{2}(D)}\;\leq\delta^{-h}\;\left\|u\right\|_{\mathbb{L}^{2}(D)}.$

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# Semi-discrete Grüss-Voronovskaya-type and Grüss-type estimates for
Bernstein-Kantorovich polynomials

Sorin G. Gal
University of Oradea
Department of Mathematics and Computer Science
Str. Universitatii Nr. 1
410087 Oradea, Romania
e-mail<EMAIL_ADDRESS>
Abstract. The aim of this note is to prove a semi-discrete Grüss-Voronovskaya-
type estimate for Bernstein-Kantorovich polynomials. Also, as a consequence, a
perturbed Grüss-type estimate is obtained.

Keywords. Bernstein-Kantorovich polynomials, semi-discrete Grüss-Voronovskaya-
type estimate, perturbed Grüss-type estimate, modulus of continuity.

AMS 2000 Mathematics Subject Classification: 41A36, 41A25, 41A60.

##  1 Introduction

A classical result in approximation theory is the asymptotic qualitative
result of Voronovskaya for Bernstein polynomials in [11]. It was generalized
by Bernstein in [3] and then it was extended to positive and linear operators
by Mamedov in [10]. Also, quantitative estimates of Mamedov’s result were
obtained in terms of the least concave majorant and a $K$-functional by Gonska
in [7] and by Gavrea-Ivan in [6].

Another classical result is the well-known Grüss inequality for positive
linear functionals $L:C[0,1]\to\mathbb{R}$. This inequality gives an upper
bound for the generalized Chebyshev functional

$T(f,g):=L(f\cdot g)-L(f)\cdot L(g),\quad f,g\in C[0,1].$

For positive and linear operators $H:C[0,1]\to C[0,1]$ reproducing constant
functions, this was investigated for the first time in [2], then obtaining in
[8] the estimate

$|H(fg;x)-H(f;x)\cdot H(g;x)|$
$\leq\frac{1}{4}\cdot\tilde{\omega}_{1}(f;2\cdot\sqrt{H(e_{2};x)-H(e_{1};x)^{2}})\cdot\tilde{\omega}_{1}(g;2\cdot\sqrt{H(e_{2};x)-H(e_{1};x)^{2}})$

where $\tilde{\omega}_{1}$ is the least concave majorant of $\omega_{1}$ and
$e_{i}(x)=x^{i}$ for $x\in[0,1]$.

A mixture between the above two classical results are the so-called Grüss-
Voronovskaya-type results obtained for the first time in the paper [5] for
Bernstein and Pǎltǎnea operators.

On the other hand, in the very recent paper [4], we generalized the asymptotic
quantitative Voronovskaya-type results, by obtaining semi-discrete
quantitative Voronovskaya-type results for general positive and linear
operators.

The main goal of this short note is to use the result in [4] to obtain in
Section 2 a semi-discrete Grüss-Voronovskaya-type results for the Bernstein-
Kantorovich polynomials. Also, as a consequence, we easily obtain a perturbed
Grüss-type estimate for the same polynomials.

##  2 Semi-discrete Grüss-Voronovskaya-type estimate

The main result is the following semi-discrete Grüss-Voronovskaya-type
estimate, for the Bernstein-Kantorovich polynomials given by the formula (see
[9])

$K_{n}(f)(x)=\sum_{k=0}^{n}p_{n,k}(x)\cdot(n+1)\int_{k/(n+1)}^{(k+1)/(n+1)}f(t)dt,$

where $p_{n,k}(x){n\choose k}x^{k}(1-x)^{n-k}$ and $f:[0,1]\to\mathbb{R}$ is
Riemann (or Lebesgue) integrable on $[0,1]$.

Also, for $n\in\mathbb{N}$ and $,y\in[0,1]$, let us denote

$\begin{split}E_{n}(x,y)&=\frac{1}{(n+1)^{2}}\cdot\left(x(1-x)(n-1)+\frac{1}{3}\right)+(x-y)\frac{1-2x}{2(n+1)}\\\
F_{n}(x)&=\frac{1}{(n+1)^{2}}\cdot\left(x(1-x)(n-1)+\frac{1}{3}\right).\end{split}$

Notice that clearly we have
$|E_{n}(x,y)|={\mathcal{O}}\left(\frac{1}{n}\right)$ and
$|F_{n}(x)|={\mathcal{O}}\left(\frac{1}{n}\right)$, uniformly with respect to
$x,y\in[0,1]$.

######  Theorem 2.1.

For all $f,g\in C^{2}[0,1]$, $n\in\mathbb{N}$ and $x,y\in[0,1]$, $x\not=y$ we
have

$\begin{split}&\left|K_{n}(fg)(x)-K_{n}(f)(x)\cdot
K_{n}(g)(x)+(x-y)\cdot\frac{1-2x}{2(n+1)}\left([x,y;f]\cdot[x,y;g]-f^{\prime}(x)g^{\prime}(x)\right)\right.\\\
-&F_{n}(x)\cdot f^{\prime}(x)\cdot g^{\prime}(x)|\\\
\leq&\left[\frac{1}{(n+1)^{2}}(x(1-x)(n-1)+1/3)+|x-y|\cdot\frac{1}{\sqrt{3}\sqrt{n+1}}\right]\\\
\cdot&\left[\omega_{1}\left((fg)^{\prime\prime};|x-y|+\frac{2\sqrt{6}}{\sqrt{n+1}}\right)+\|g\|\omega_{1}\left(f^{\prime\prime};|x-y|+\frac{2\sqrt{6}}{\sqrt{n+1}}\right)\right.\\\
&\left.+\|f\|\omega_{1}\left(g^{\prime\prime};|x-y|+\frac{2\sqrt{6}}{\sqrt{n+1}}\right)\right]+|K_{n}(f)(x)-f(x)|\cdot|K_{n}(g)(x)-g(x)|,\end{split}$

where $[x,y;f]=\frac{f(x)-f(y)}{x-y}$,
$\omega_{1}(f;\delta):=\sup\\{|f(x)-f(y)|;x,y\in[0,1],|x-y|\leq\delta\\}$ and
$\|f\|$ denotes the uniform norm of $f$.

###### Proof.

Supposing that $f,g\in C^{2}[0,1]$ and using Corollary 4.1 in [4], we obtain

$\begin{split}&\left|K_{n}(fg)(x)-K_{n}(f)(x)\cdot
K_{n}(g)(x)+(x-y)\cdot\frac{1-2x}{2(n+1)}\left([x,y;f]\cdot[x,y;g]-f^{\prime}(x)g^{\prime}(x)\right)\right.\\\
-&F_{n}(x)\cdot f^{\prime}(x)\cdot g^{\prime}(x)|\\\
&=\left|\left(K_{n}(fg)(x)-f(x)g(x)-\frac{1-2x}{2(n+1)}\cdot[x,y;fg]\right)-E_{n}(x,y)\cdot\frac{(f(x)g(x))^{\prime\prime}}{2}\right.\\\
&\left.-g(x)\left[\left(K_{n}(f)(x)-f(x)-\frac{1-2x}{2(n+1)}\cdot[x,y;f]\right)-E_{n}(x,y)\cdot\frac{f^{\prime\prime}(x)}{2}\right]\right.\\\
&\left.-f(x)\left[\left(K_{n}(g)(x)-g(x)-\frac{1-2x}{2(n+1)}\cdot[x,y;g]\right)-E_{n}(x,y)\cdot\frac{g^{\prime\prime}(x)}{2}\right]\right.\\\
&+\left.[K_{n}(f)(x)-f(x)]\cdot[g(x)-K_{n}(g)(x)]\right|\\\
&\leq\left[\frac{1}{(n+1)^{2}}(x(1-x)(n-1)+1/3)+|x-y|\cdot\frac{1}{\sqrt{3}\sqrt{n+1}}\right]\\\
&\cdot\left[\omega_{1}\left((fg)^{\prime\prime};|x-y|+\frac{2\sqrt{6}}{\sqrt{n+1}}\right)+\|g\|\cdot\omega_{1}\left(f^{\prime\prime};|x-y|+\frac{2\sqrt{6}}{\sqrt{n+1}}\right)\right.\\\
&+\left.\|f\|\cdot\omega_{1}\left(g^{\prime\prime};|x-y|+\frac{2\sqrt{6}}{\sqrt{n+1}}\right)\right]+|K_{n}(f)(x)-f(x)|\cdot|K_{n}(g)(x)-g(x)|,\end{split}$

which is exactly the estimate in the statement. ∎

######  Remark 2.2.

Let $f,g\in C^{3}[0,1]$. Firstly, take $y\to x$, multiply by $n$ both members
in the estimate in Theorem 2.1 and use the estimate in [1], page 849, line 7
from below

$|K_{n}(h)(x)-h(x)|\leq\frac{1}{2n}\|h^{\prime}\|+\frac{8}{9n}\|h^{\prime\prime}\|,x\in[0,1],n\in\mathbb{N},h\in
C^{2}[0,1].$

Then, since

$n\cdot F_{n}(x)=\frac{n(n-1)}{(n+1)^{2}}x(1-x)+\frac{n}{3(n+1)^{2}},$

by using the estimate in Theorem 2.1, we easily obtain

$\|n[K_{n}(fg)-K_{n}(f)\cdot
K_{n}(g)]-e_{1}(1-e_{1})f^{\prime}g^{\prime}\|={\mathcal{O}}\left(\frac{1}{\sqrt{n}}\right),$

thus recapturing the order of approximation in the classical Grüss-
Voronovskaya-type estimate given by Theorem 5.1 in [1].

######  Remark 2.3.

Since obviously

$\left|K_{n}(fg)(x)-K_{n}(f)(x)\cdot
K_{n}(g)(x)+(x-y)\cdot\frac{1-2x}{2(n+1)}\left([x,y;f]\cdot[x,y;g]-f^{\prime}(x)g^{\prime}(x)\right)\right|$
$\begin{split}&\leq\left|K_{n}(fg)(x)-K_{n}(f)(x)\cdot
K_{n}(g)(x)+(x-y)\cdot\frac{1-2x}{2(n+1)}\left([x,y;f]\cdot[x,y;g]-f^{\prime}(x)g^{\prime}(x)\right)\right.\\\
&\left.-F_{n}(x)\cdot f^{\prime}(x)\cdot
g^{\prime}(x)\right|+|F_{n}(x)|\cdot\left|f^{\prime}(x)\cdot
g^{\prime}(x)\right|,\end{split}$

by Theorem 2.1, for all $f,g\in C^{2}[0,1]$, $n\in\mathbb{N}$, $x,y\in[0,1]$,
$x\not=y$, we immediately get the following estimate

$\begin{split}&\left|K_{n}(fg)(x)-K_{n}(f)(x)\cdot
K_{n}(g)(x)+(x-y)\cdot\frac{1-2x}{2(n+1)}\left([x,y;f]\cdot[x,y;g]-f^{\prime}(x)g^{\prime}(x)\right)\right|\\\
&\leq\left[\frac{1}{(n+1)^{2}}(x(1-x)(n-1)+1/3)+|x-y|\cdot\frac{1}{\sqrt{3}\sqrt{n+1}}\right]\\\
&\cdot\left[\omega_{1}\left((fg)^{\prime\prime};|x-y|+\frac{2\sqrt{6}}{\sqrt{n+1}}\right)+\|g\|\cdot\omega_{1}\left(f^{\prime\prime};|x-y|+\frac{2\sqrt{6}}{\sqrt{n+1}}\right)\right.\\\
&+\left.\|f\|\cdot\omega_{1}\left(g^{\prime\prime};|x-y|+\frac{2\sqrt{6}}{\sqrt{n+1}}\right)\right]+|K_{n}(f)(x)-f(x)|\cdot|K_{n}(g)(x)-g(x)|\\\
&+|F_{n}(x)|\cdot|f^{\prime}(x)g^{\prime}(x)|,\end{split}$

which can be considered as a ”perturbed” (discrete) Grüss-type estimate since
for $y$ sufficiently close to $x$, the left-hand side of the above inequality,
becomes sufficiently close to

$|K_{n}(fg)(x)-K_{n}(f)(x)\cdot K_{n}(g)(x)|.$

Now, if above we take $y\to x$, then since
$|F_{n}(x)|={\mathcal{O}}\left(\frac{1}{n}\right)$, we immediately get

$\|K_{n}(fg)-K_{n}(f)\cdot K_{n}(g)\|={\mathcal{O}}\left(\frac{1}{n}\right),$

which is the same order which can be obtained for the classical Grüss-type
estimate in terms of the least concave majorant of the modulus of continuity
expressed by Theorem 4.2 in [1] for $f,g\in C^{2}[0,1]$.

######  Remark 2.4.

The results in this note suggest that based on other semi-discrete
Voronovskaya-type results in [4], to get for the Bernstein-Kantorovich
polynomials other semi-discrete estimates of Grüss-Voronovskaya-type and of
Grüss-type. Also, similar results can be obtained for other positive and
linear operators too.

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