Bochner-Riesz Means Convergence of Prolate Spheroidal Series and Their
Extensions

Mourad Boulsanea 111 Corresponding author: Mourad Boulsane, Email:
<EMAIL_ADDRESS>Ahmed Souabnia

a Carthage University, Faculty of Sciences of Bizerte, Department of
Mathematics, Jarzouna, 7021, Tunisia.

###### Abstract

In this paper, we study the $L^{p}$-Bochner-Riesz mean summability problem
related to the spectrum of some particular Sturm-Liouville operators in the
weighted $L^{p}([a,b],\omega).$ Our purpose is to establish suitable
conditions under which the Bochner-Riesz expansion of a function $f\in
L^{p}([a,b],\omega)$,$1<p<\infty$, in two generalisations of Slepian’s basis,
converges to $f$ in $L^{p}([a,b],\omega)$.

Keywords: Bochner-Riesz mean convergence, eigenfunctions and eigenvalues,
prolate spheroidal wave functions.
2010 Mathematics Subject Classification. 42C10, 41A60.

##  1 Introduction

The $L^{p}$-Bochner-Riesz mean convergence of orthogonal series has attracted
special attention since several decades ago. This kind of convergence is
briefly described as follows. Let $1\leq p<\infty$ , $a,b\in\mathbb{R}$ and
$\\{\varphi_{n}\\}$ an orthonormal set of eigenfunctions of a positive self-
adjoint differential operator $\mathcal{L}$ associated with eigenvalues
$\chi_{n}$ on a weighted Hilbert space $L^{2}(I,\omega)$, where $\omega$ is a
positive bounded weight function. We define the expansion coefficients of
$f\in L^{p}([a,b],\omega)$ by
$a_{n}(f)=\int_{a}^{b}f(x)\varphi_{n}(x)\omega(x)dx.$ The orthonormal set
$\\{\varphi_{n}\\}$ is said to have the Bochner-Riesz mean convergence of
order $p$ over the Banach space $L^{p}(I,\omega)$ if for some suitable
$\delta>0$ and for all $f\in L^{p}(I,\omega),$ we have

$\lim_{R\to\infty}\int_{a}^{b}|f(x)-\Psi_{R}^{\delta}f(x)|^{p}\omega(x)dx=0,\mbox{
where
}\displaystyle\Psi_{R}^{\delta}f=\sum_{n=0}^{\infty}\Big{(}1-\frac{\chi_{n}}{R}\Big{)}^{\delta}_{+}a_{n}(f)\varphi_{n}.$
(1)

To the best of our knowledge, M. Riesz was the first, in 1911, to investigate
this problem in some special cases. Our problem is a modified summability
method of Riesz mean introduced by Salomon Bochner given by (1) . In [7],
S.Bochner started by studying this problem for the trigonometric exponential
case in higher dimension. Furthermore, in [13], the authors have proved a
Bochner-Riesz mean convergence for the orthonormal eigenvectors system of a
second order elliptic differential operator on a compact N-dimensional
manifold M for $1\leq p\leq 2\frac{N+1}{N+3}$ and
$\delta>N\left|\frac{1}{p}-\frac{1}{2}\right|-\frac{1}{2}$. Mauceri and Müller
have also studied this problem in [20] and [22] in the framework of the
Heisenberg group. This problem has been analysed for Fourier-Bessel expansions
series in [10] and [11]. Moreover, in [8], authors have also solved this
question in the case of sublaplacien on the sphere $S^{2n-1}$ in the complex
n-dimensional space $\mathbb{C}^{n}$,where it has been shown that we have
convergence for $\delta>(2n-1)\Big{|}\frac{1}{2}-\frac{1}{p}\Big{|}$. The weak
type convergence is investigated in this problem. Indeed, we say that an
orthonormal family $\\{\varphi_{n}\\}$ of $L^{p}(I,\omega)$ has a weakly
Bochner-Riesz mean convergence if $\Psi_{R}^{\delta}f$ converge to $f$ almost
everywhere for every $f\in L^{p}(I,\omega)$. This problem has been solved in
some special cases of orthonormal systems like Jacobi and Laguerre polynomials
in [21] and for the eigenfunctions of the Hermite operator in higher dimension
in [9].
In this work, we extend the $L^{p}$-Bochner-Riesz means convergence to the
circular and the generalized (or weighted) prolate spheroidal wave functions
denoted by (CPSWFs) and (GPSWFs), respectively. The two last families are
defined respectively as the eigenfunctions of the operators

$\mathcal{H}_{c}^{\alpha}f(x)=\int_{0}^{1}\sqrt{cxy}J_{\alpha}(cxy)f(y)dy,\quad\mathcal{F}_{c}^{(\alpha)}f(x)=\int_{-1}^{1}e^{icxy}f(y)(1-y^{2})^{\alpha}\,dy,$

where $\alpha>-1/2,\,c>0$ are two real numbers. These two sets of orthonormal
functions are characterized as solutions of some Sturm-Liouville problems. The
second family we consider is the weighted, some times called generalized,
prolate spheroidal wave functions (GPSWFs) introduced by Wang-Zhang [30]. Note
that the classical PSWFs correspond to the special case of the GPSWFs with
$\alpha=0.$

Our aim in this paper is to prove the $L^{p}$convergence of Bochner-Riesz mean
expansion in the GPSWFs and CPSWFs bases.

This work is organised as follows. In section 2, we give some mathematical
preliminaries on Sturm-Liouville theory and some properties of the CPSWfs and
GPSWFs. Note that these functions can be considered as generalizations of the
spherical Bessel functions $j_{n}^{(\alpha)}$ and Gegenbauer’s polynomials
$\widetilde{P}_{n}^{(\alpha)}$, respectively. In section 3, we state our two
main theorems and section 4 and 5 are respectively devoted to the proof of
sufficient and necessary conditions of the main results.

##  2 Mathematical preliminaries

In this paragraph, we give some mathematical preliminaries that will be
frequently used in the proofs of the different results of this work.

###  2.1 Some facts about Sturm-Liouville theory

The Sturm-Liouville differential operator is defined as follows, see for
example [1],

$\mathcal{L}y(x)=\frac{d}{dx}[p(x)y^{\prime}(x)]+q(x)y(x),\quad x\in I=(a,b).$
(2)

with $r=\frac{1}{p},q\in L^{1}(I,\mathbb{R}).$ The Sturm-Liouville eigenvalues
problem is given by the following differential equation :

$\mathcal{L}.u(x)=-\chi\omega(x)u(x),\quad\sigma\in L^{1}(I,\mathbb{R}).$ (3)

That is

$\frac{d}{dx}\Big{[}p(x)\frac{du}{dx}\Big{]}+q(x)u(x)+\chi\omega(x)u(x)=0,\quad
x\in I.$ (4)

Note that a Sturm-Liouville operator satisfies the following properties,

  1. 1.

$u\mathcal{L}v-v\mathcal{L}u=\Big{[}p(uv^{\prime}-vu^{\prime})\Big{]}^{\prime}$
( Lagrange’s identity )

  2. 2.

The eigenvalues of $\mathcal{L}$ are real and form an infinite countable set
$\chi_{0}<\chi_{1}<\cdots<\chi_{n}<\cdots$ with
$\lim_{n\rightarrow+\infty}\chi_{n}=+\infty.$

  3. 3.

For each eigenvalue $\chi_{n}$ there exists an eigenfunction $\phi_{n}$ having
n zeros on $[a,b].$

  4. 4.

Eigenfunctions corresponding to different eigenvalues are orthogonal with
respect to the following inner product

${\left\langle{f,g}\right\rangle}_{\omega}=\int_{a}^{b}f(x)g(x)\omega(x)dx,\quad
f,g\in L^{2}(I,\omega).$

In the sequel, we assume that $\omega(x)\geq 0$, for $x\in(a,b).$

###  2.2 Some facts about GPSWFs and CPSWFs

We first recall that, for $c>0$, the prolate spheroidal wave functions PSWFs,
denoted $\psi_{n,c}$, have been introduced by D.Slepian as solutions of the
following energy maximization problem

$\mbox{ Find }f=\arg\max_{f\in
B_{c}}\frac{\int_{-1}^{1}|f(t)|^{2}dt}{\int_{\mathbb{R}}|f(t)|^{2}dt},$

where $B_{c}$ is the classical Paley-Wiener space, defined by

$B_{c}=\left\\{f\in L^{2}(\mathbb{R}),\,\,\mbox{Support
}\widehat{f}\subseteq[-c,c]\right\\}.$ (5)

Here, $\widehat{f}$ is the Fourier transform of $f\in L^{2}(\mathbb{R}).$ It
has been shown that they are also eigenfunctions of the integral operator with
sinc kernel. A breakthrough in the theory of Slepian functions is due to
Slepian,Pollard and Landau who have proved that PSWFs are also eigenfunctions
of a Sturm-Liouville operator by proving a commutativity property. For more
details about Slepian’s functions we refer reader to [25, 26, 27]. In this
work we are interested in two generalizations of the PSWFs.
The first basis is called circular prolate spheroidal wave functions (CPSWFs)
or radial part of the 2d-Slepian, introduced by D.Slepian[27] as solutions of
the following problem

$\mbox{ Find }f=\arg\max_{f\in
HB^{\alpha}_{c}}\frac{\int_{0}^{1}|f(t)|^{2}dt}{\int_{0}^{\infty}|f(t)|^{2}dt},$

where $HB^{\alpha}_{c}$ is the Hankel Paley-Wiener space, defined by

$HB^{\alpha}_{c}=\left\\{f\in L^{2}(\mathbb{R}),\,\,\mbox{Support
}\mathcal{H}^{\alpha}f\subseteq[-c,c]\right\\}.$ (6)

Here the Hankel transform $\mathcal{H}^{\alpha}$ is defined, for $f\in
L^{1}(0,\infty)$, by

$\mathcal{H}^{\alpha}f(x)=\int_{0}^{\infty}\sqrt{xy}J_{\alpha}(xy)f(y)dy.$

Here $J_{\alpha}(.)$ is the Bessel function and $\alpha>-1/2$. Like Fourier
transform, $\mathcal{H}^{\alpha}$ can be extended into a unitary operator on
$L^{2}(0,\infty)$. They are also the different band-limited eigenfunctions of
the finite Hankel transform $\mathcal{H}_{c}^{\alpha}$ defined on $L^{2}(0,1)$
with kernel $H_{c}^{\alpha}(x,y)=\sqrt{cxy}J_{\alpha}(cxy)$ where $J_{\alpha}$
is the Bessel function of the first type and order $\alpha>-\frac{1}{2}$(see
for example [27]). That is

$\mathcal{H}_{c}^{\alpha}(\varphi^{\alpha}_{n,c})=\mu_{n,\alpha}(c)\varphi^{\alpha}_{n,c}.$
(7)

In his pioneer work [27], D. Slepian has shown that the compact integral
operator $\mathcal{H}_{c}^{\alpha}$ commutes with the following Sturm-
Liouville differential operator $\mathcal{L}^{\alpha}_{c}$ defined on
$C^{2}([0,1])$ by

$\mathcal{L}_{c}^{\alpha}(\phi)=-\dfrac{d}{dx}\left[(1-x^{2})\dfrac{d}{dx}\phi\right]+\left(c^{2}x^{2}-\dfrac{\dfrac{1}{4}-\alpha^{2}}{x^{2}}\right)\phi.$
(8)

Hence, $\varphi^{\alpha}_{n,c}$ is the $n-$th bounded eigenfunction of the
positive self-adjoint operator $\mathcal{L}_{c}^{\alpha}$ associated with the
eigenvalue $\chi_{n,\alpha}(c),$ that is

$-\dfrac{d}{dx}\left[(1-x^{2})\dfrac{d}{dx}\varphi^{\alpha}_{n,c}(x)\right]+\left(c^{2}x^{2}-\dfrac{\dfrac{1}{4}-\alpha^{2}}{x^{2}}\right)\varphi^{\alpha}_{n,c}(x)=\chi_{n,\alpha}(c)\varphi^{\alpha}_{n,c}(x),\quad
x\in[0,1].$ (9)

The orthonormal family $\varphi_{n,c}^{\alpha}$ form an orthonormal basis of
$L^{2}(0,1)$ and the associated eigenvalues family $\chi_{n,\alpha}(c)$
satisfy the following inequality, (see [27])

$(2n+\alpha+1/2)(2n+\alpha+3/2)\leq\chi_{n,\alpha}(c)\leq(2n+\alpha+1/2)(2n+\alpha+3/2)+c^{2}$
(10)

The second family we consider in this work is the weighted, (some times called
generalized), prolate spheroidal wave functions introduced by Wang-Zhang [30]
as solutions of a Sturm-Liouville problem or equivalently eigenfunctions of an
integral operator. GPSWFs are also solutions of the following problem as given
in [18]

$\mbox{Find }f={\displaystyle arg\max_{f\in
B^{\alpha}_{c}}\frac{\|f\|^{2}_{L^{2}_{\omega_{\alpha}}(I)}}{\|\widehat{f}\|^{2}_{L^{2}(\omega_{-\alpha}(\frac{\cdot}{c}))}}},$

where $\omega_{\alpha}(x)=(1-x^{2})^{\alpha}$ and $B^{(\alpha)}_{c}$ is the
restricted Paley-Winer space, defined by

$B_{c}^{(\alpha)}=\\{f\in L^{2}(\mathbb{R}),\,\,\mbox{Support
}\widehat{f}\subseteq[-c,c],\,\,\widehat{f}\in
L^{2}\big{(}(-c,c),\omega_{-\alpha}(\frac{\cdot}{c})\big{)}\\}.$

More precisely, the GPSWFs are the eigenfunctions of the weighted finite
Fourier transform operator $\mathcal{F}_{c}^{(\alpha)}$ defined by

$\mathcal{F}_{c}^{(\alpha)}f(x)=\int_{-1}^{1}e^{icxy}f(y)\,\omega_{\alpha}(y)\,\mathrm{d}y.$
(11)

It is well known, (see [18, 30]) that they are also eigenfunctions of the
compact and positive operator

$\mathcal{Q}_{c}^{(\alpha)}=\frac{c}{2\pi}\mathcal{F}_{c}^{({\alpha})^{*}}\circ\mathcal{F}_{c}^{(\alpha)}$

which is defined on $L^{2}(I,\omega_{\alpha})$ by

$\mathcal{Q}_{c}^{(\alpha)}g(x)=\int_{-1}^{1}\frac{c}{2\pi}\mathcal{K}_{\alpha}(c(x-y))g(y)\omega_{\alpha}(y)\d{y}$
(12)

Here,

$\mathcal{K}_{\alpha}(x)=\sqrt{\pi}2^{\alpha+1/2}\Gamma(\alpha+1)\frac{J_{\alpha+1/2}(x)}{x^{\alpha+1/2}}$

It has been shown in [18, 30] that the last two integral operators commute
with the following Sturm-Liouville operator $\mathcal{L}_{c}^{(\alpha)}$
defined on $C^{2}[-1,1]$ by

$\mathcal{L}_{c}^{(\alpha)}(f)(x)=-\frac{1}{\omega_{\alpha}(x)}\frac{d}{dx}\left[\omega_{\alpha}(x)(1-x^{2})f^{\prime}(x)\right]+c^{2}x^{2}f(x).$
(13)

Also, note that the $(n+1)-$th eigenvalue $\chi_{n,\alpha}(c)$ of
$\mathcal{L}_{c}^{(\alpha)}$ satisfies the following classical inequalities,

$n(n+2\alpha+1)\leq\chi_{n,\alpha}(c)\leq n(n+2\alpha+1)+c^{2},\quad\forall
n\geq 0.$ (14)

##  3 Statement of results

In this section, we will state the main results of this paper that we will
prove in the following sections. As mentioned before, the main issue studied
in this paper is to get a necessary and sufficient conditions of Bochner-Riesz
expansion convergence of a function $f$ in the GPSWFs’s and CPSWFs’s basis.
Let’s start by studying the case of GPSWFs in the following theorem.

######  Theorem 1.

Let $0\leq\alpha<3/2$, $\delta$ and $c$ be two positive number and
$(\psi_{n,c}^{(\alpha)})_{n\geq 0}$ be the family of weighted prolate
spheroidal wave functions. For a smooth function $f$ on $I=(-1,1)$, we define

$\Psi_{R}^{\delta}f=\sum_{n=0}^{\infty}\left(1-\frac{\chi_{n,\alpha}(c)}{R}\right)^{\delta}_{+}{\left\langle{f,\psi_{n,c}^{(\alpha)}}\right\rangle}_{L^{2}(I,\omega_{\alpha})}\psi_{n,c}^{(\alpha)}.$

Then, for every $1\leq p<\infty$, $\Psi^{\delta}_{R}$ can be extended to a
bounded operator $L^{p}(I,\omega_{\alpha})\to L^{p}(I,\omega_{\alpha})$.
Further, $\Psi^{\delta}_{R}f$ is uniformly bounded if ,and only if,
$\delta>\max\\{\frac{\gamma_{\alpha}(p^{\prime})}{2},0\\}$ and
$p\not=p_{0}=2-\frac{1}{\alpha+3/2}$ where

$\gamma_{\alpha}(p)=\begin{cases}0&\mbox{ if }1<p<p^{\prime}_{0}\\\
\epsilon&\mbox{ if }p=p^{\prime}_{0}\\\
2(\alpha+1)\left[\frac{1}{2}-\frac{1}{p}\right]-\frac{1}{2}&\mbox{ if
}p>p^{\prime}_{0}\\\ \alpha+1&\mbox{ if }p=1\end{cases}.$

and $\epsilon$ is an arbitrary real number. Note that $p^{\prime}$ denote here
the dual exponent of $p$.

######  Remark 1.

The sufficient condition in both GPSWFs’s and CPSWFs’s case still valid even
for all $\alpha>-1/2$.

######  Remark 2 (Two special cases).

Recall that $\psi^{(\alpha)}_{n,0}=\widetilde{P}^{(\alpha,\alpha)}_{n}$, then
we recover the same result for the case of normalized Geganbauer polynomials.
Note that both conditions (A) and (B), defined in the proof of the last
theorem, are still valid even for $\widetilde{P}_{n}^{(\alpha,\beta)}$ with
exactly the same proof and by noticing that the transferring theorem, which is
the key step of the necessary condition, has been proven in [16] , the last
result is valid also for Jacobi polynomials for all $\alpha,\beta>-1/2$ . For
$\alpha=0$ and $c>0$, $\psi^{0}_{n,c}=\psi_{n,c}$ presents the classical
prolate spheroidal wave functions PSWFs which satisfy the Bochner-Riez mean
convergence if and only if
$\delta>\max\\{0,\frac{\gamma_{0}(p^{\prime})}{2}\\}.$

Let us now focus on the circular case.

######  Theorem 2.

Let $\alpha\geq 1/2$, $c>0$ and $(\varphi_{n,c}^{(\alpha)})_{n\geq 0}$ be the
family of Hankel prolate spheroidal wave functions. For a smooth function $f$
on $I=(0,1)$, we define

$\Psi_{R}^{\delta}f=\sum_{n=0}^{\infty}\left(1-\frac{\chi_{n,\alpha}(c)}{R}\right)^{\delta}_{+}{\left\langle{f,\varphi_{n,c}^{(\alpha)}}\right\rangle}_{L^{2}(0,1)}\varphi_{n,c}^{(\alpha)}.$

Then, for every $1\leq p<\infty$, $\Psi^{\delta}_{R}$ can be extended to a
bounded operator $L^{p}(0,1)\to L^{p}(0,1)$. Further $\Psi^{\delta}_{R}f$ is
uniformly bounded if ,and only if,
$\delta>\max\\{\frac{\gamma(p^{\prime})}{2},0\\}$ where

$\gamma(p)=\begin{cases}\frac{1}{p}-\frac{1}{2}&\mbox{ if }1<p<4\\\
\epsilon-\frac{1}{4}&\mbox{ if }p=4\\\
\frac{1}{3}\left[\frac{1}{p}-1\right]&\mbox{ if }p>4\\\ 1&\mbox{ if
}p=1\end{cases}.$

.

##  4 Proof of sufficient condition

Let $(I,\omega)$ be a measured space such that $\omega$ is a bounded weight
function. We denote by $p^{\prime}=\frac{p}{p-1}$ the dual index of $p$.

Throughout this section, $\mathcal{L}$ denotes a Sturm-Liouville operator and
$\varphi_{n}$ (respectively $\lambda_{n}$) the sequence of the associated
eigenfunctions (respectively eigenvalues). The Riesz means of index $\delta>0$
associated with $\mathcal{L}$ of a function
$f\in\mathcal{C}^{\infty}(I,\mathbb{R})$ are consequently defined as

$\Psi^{\delta}_{R}f=\sum_{n=0}^{\infty}\Big{(}1-\frac{\lambda_{n}}{R}\Big{)}^{\delta}_{+}a_{n}(f)\varphi_{n}\quad\mbox{with}\quad
a_{n}(f)=\int_{I}f(y)\varphi_{n}(y)d\mu(y).$ (15)

$\Psi^{\delta}_{R}f$ can also be written as

$\Psi^{\delta}_{R}.f(x)=\int_{I}K_{R}^{\delta}(x,y)f(y)d\mu(y)\quad\mbox{where}\quad
K_{R}^{\delta}(x,y)=\sum_{n=0}^{\infty}\Big{(}1-\frac{\lambda_{n}}{R}\Big{)}^{\delta}_{+}\varphi_{n}(x)\varphi_{n}(y)$

Our aim in this section is to prove at the same time the sufficient conditions
of our two main theorems. More precisely, we will define several conditions on
$\varphi_{n}$ that will ensure the convergence of $\Psi_{R}^{\delta}.f$ to $f$
in the $L^{p}$norm as $R\to\infty$ and verify that both two families satisfy
these conditions. Assume that $\varphi_{n}$ satisfies the following conditions
:

  * $(A)$

For every $1\leq p\leq\infty$, every $n$, $\varphi_{n}\in L^{p}(I,\omega)$.
Further, we assume that there is a constant $\gamma(p)\geq 0$ such that
${\left\|{\varphi_{n}}\right\|}_{L^{p}(\mu)}\leq Cn^{\gamma(p)}$.

  * $(B)$

The sequence $(\lambda_{n})$ of the eigenvalues of the operator $\mathcal{L}$
satisfies the following properties

    1. 1.

$\displaystyle\sum_{\lambda_{n}\in(m,M)}1\leq C(M-m)$ for all $0\leq m<M$.

    2. 2.

There exists $\varepsilon>0$ such that

$\lambda_{n}\geq Cn^{\varepsilon}.$

First of all, we start by giving sense to $\Psi^{\delta}_{R}.f$ for every
$f\in L^{p}(\mu)$. Indeed,

${\left\|{K^{\delta}_{R}}\right\|}_{L^{p}(\mu)\otimes
L^{p^{\prime}}(\mu)}\leq\sum_{\lambda_{n}<R}{\left\|{\varphi_{n}}\right\|}_{p}{\left\|{\varphi_{n}}\right\|}_{p^{\prime}}\leq\sum_{\lambda_{n}<R}n^{\gamma(p)+\gamma(p^{\prime})}\leq
CR^{\frac{\left(\gamma(p)+\gamma(p^{\prime})\right)}{\varepsilon}+1},$

So that the integral operator $\Psi_{R}^{\delta}$ can be extended to a
continuous operator $L^{p}(\mu)\to L^{p}(\mu)$ with

${\left\|{\Psi_{R}^{\delta}}\right\|}_{L^{p}\to
L^{p}}\leq{\left\|{K^{\delta}_{R}}\right\|}_{L^{p}(\mu)\otimes
L^{p^{\prime}}(\mu)}.$

The following theorem is one of the main results of this paper.

######  Theorem 3.

With the above notation and under conditions $(A)$ and $(B)$ with
$\delta>\delta(p)=\max\\{\frac{\gamma(p^{\prime})}{\varepsilon},0\\}$, there
exists a constant $C>0$ satisfying the following inequality

${\left\|{\Psi_{R}^{\delta}}\right\|}_{(L^{p}(I,w),L^{p}(I,w))}\leq C.$ (16)

The following lemma will be used in the proof of the previous theorem.

######  Lemma 1.

Let $1\leq p\leq 2$ then for every $f\in L^{p}(I,\omega),$ we have

${\left\|{\sum_{\lambda_{n}\in(m,M)}a_{n}(f)\varphi_{n}}\right\|}_{L^{2}(I,\omega)}\leq
C(p)M^{\frac{\gamma(p^{\prime})}{\varepsilon}}(M-m)^{\frac{1}{2}}{\left\|{f}\right\|}_{L^{p}(I,\omega)}.$
(17)

###### Proof.

By orthogonality and Hölder’s inequality, we have

$\displaystyle{\left\|{\sum_{\lambda_{n}\in(m,M)}a_{n}(f)\varphi_{n}}\right\|}^{2}_{L^{2}(I,\omega)}$
$\displaystyle=$
$\displaystyle\sum_{\lambda_{n}\in(m,M)}a_{n}^{2}(f)\leq\sum_{\lambda_{n}\in(m,M)}{\left\|{\varphi_{n}}\right\|}^{2}_{L^{p^{\prime}}(I,\omega)}{\left\|{f}\right\|}^{2}_{L^{p}(I,\omega)}$

From condition $(A)$, we have
${\left\|{\varphi_{n}}\right\|}_{L^{p^{\prime}}(I,\omega)}\leq
n^{\gamma(p^{\prime})}$. We also obtain by using condition ($B1$)

$\displaystyle{\left\|{\sum_{\lambda_{n}\in(m,M)}a_{n}(f)\varphi_{n}}\right\|}^{2}_{L^{2}(I,\omega)}$
$\displaystyle\leq$
$\displaystyle\sum_{\lambda_{n}\in(m,M)}n^{2\gamma(p^{\prime})}{\left\|{f}\right\|}^{2}_{L^{p}(I,\omega)}\leq
C\sum_{\lambda_{n}\in(m,M)}\lambda_{n}^{\frac{2\gamma(p^{\prime})}{\varepsilon}}{\left\|{f}\right\|}^{2}_{L^{p}(I,\omega)}$
$\displaystyle\leq$ $\displaystyle
CM^{\frac{2\gamma(p^{\prime})}{\varepsilon}}\left(\sum_{\lambda_{n}\in(m,M)}1\right){\left\|{f}\right\|}^{2}_{L^{p}(I,\omega)}$
$\displaystyle\leq$ $\displaystyle
CM^{(\frac{2\gamma(p^{\prime})}{\varepsilon})}(M-m){\left\|{f}\right\|}^{2}_{L^{p}(I,\omega)}.$

Then one gets

${\left\|{\sum_{\lambda_{n}\in(m,M)}a_{n}(f)\varphi_{n}}\right\|}_{L^{2}(I,\omega)}\leq
CM^{(\frac{\gamma(p^{\prime})}{\varepsilon})}(M-m)^{\frac{1}{2}}{\left\|{f}\right\|}_{L^{p}(I,\omega)}.$

∎

###### Proof of Theorem1.

We should mention here that some parts of the proof of this theorem are
inspired from [8]. Without loss of generality, we can consider $1\leq p<2$ and
conclude by duality. To prove (16), we are going to have to decompose the
multiplier $\Psi_{R}^{\delta}$. In order to do so, let
$\phi\in\mathcal{C}^{\infty}_{0}(\mathbb{R})$ with support on $(1/2,2)$ such
that $\displaystyle\sum_{k\in\mathbb{Z}}\phi(2^{k}t)=1$ and
$\displaystyle\phi_{0}(t)=1-\sum_{k=1}^{+\infty}\phi(2^{k}t)$ for all $t>0$.
We define

$\phi_{R,k}^{\delta}(t)=\left(1-\frac{t}{R}\right)_{+}^{\delta}\phi\left(2^{k}(1-\frac{t}{R})\right).$

We recall that, from [8], this last function has the following properties :

  1. 1.

$\mbox{supp}\left(\phi_{R,k}^{\delta}\right)\subseteq(R(1-2^{-k+1}),R(1-2^{-k-1}))$,

  2. 2.

$\sup_{t\in\mathbb{R}}|\phi_{R,k}^{\delta}(t)|\leq C2^{-k\delta}$,

  3. 3.

$\forall N\geq 0,$ there exists $C_{N}>0$ such that

$|\partial_{t}^{N}\phi_{R,k}^{\delta}(t)|\leq
C_{N}\Big{(}\frac{2^{k}}{R}\Big{)}^{N}.$

Furthermore, we denote by

$\Psi_{R,k}^{\delta}.f=\sum_{n=0}^{\infty}\phi_{R,k}^{\delta}(\lambda_{n})a_{n}(f)\varphi_{n}\qquad
k=1,2,\cdots$ (18)

Then, we have

$\displaystyle\Psi_{R}^{\delta}f$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(1-\frac{\lambda_{n}}{R}\right)_{+}^{\delta}a_{n}(f)\varphi_{n}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\phi_{0}(1-\frac{\lambda_{n}}{R})\left(1-\frac{\lambda_{n}}{R}\right)_{+}^{\delta}a_{n}(f)\varphi_{n}+\sum_{n=0}^{\infty}\left(\sum_{k=1}^{+\infty}\phi(2^{k}(1-\frac{\lambda_{n}}{R}))\right)\left(1-\frac{\lambda_{n}}{R}\right)_{+}^{\delta}a_{n}(f)\varphi_{n}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\phi_{0}(1-\frac{\lambda_{n}}{R})\left(1-\frac{\lambda_{n}}{R}\right)_{+}^{\delta}a_{n}(f)\varphi_{n}+\sum_{k=1}^{\left[\frac{\log(R)}{\log(2)}\right]}\sum_{n=0}^{\infty}\phi_{R,k}^{\delta}(\lambda_{n})a_{n}(f)\varphi_{n}+\sum_{k=\left[\frac{\log(R)}{\log(2)}\right]+1}^{\infty}\sum_{n=0}^{\infty}\phi_{R,k}^{\delta}(\lambda_{n})a_{n}(f)\varphi_{n}$
$\displaystyle=$
$\displaystyle\psi_{R,0}^{\delta}f+\sum_{k=1}^{\left[\frac{\log(R)}{\log(2)}\right]}\Psi_{R,k}^{\delta}f+\mathcal{R}_{R}^{\delta}f.$

It is clear that the main term is the second one. With the same approach used
in [8], we will prove the following proposition :

######  Proposition 1.

Let $1\leq p<2$ and $\delta>\delta(p)=\frac{\gamma(p^{\prime})}{\varepsilon}$.
There exists $\beta>0$ such that for every $f\in L^{p}(I,w)$, we have

${\left\|{\Psi_{R,k}^{\delta}f}\right\|}_{L^{p}(I,w)}\leq
C2^{-k\beta}{\left\|{f}\right\|}_{L^{p}(I,w)},$ (19)

where $C$ is a constant independent of $R$ and $f$.

###### Proof.

Let $x_{0}=\frac{a+b}{2}\in(a,b)$ and $r=\frac{b-a}{4}>0$ such that
$(x_{0}-r,x_{0}+r)\subseteq(a,b).$ Note that, for every $1\leq
k\leq\left[\frac{\log(R)}{\log(2)}\right]=k_{R}$, we have
$r_{k}^{\alpha}=\left(\frac{2^{k}}{R}\right)^{\mu(p)}r<r$ where
$\mu(p)=\frac{(\frac{\gamma(p^{\prime})}{\varepsilon}+\frac{1}{2})}{(\frac{1}{p}-\frac{1}{2})}$
. So we notice that
$I=(a,b)=(x_{0}-r_{k}^{\alpha},x_{0}+r_{k}^{\alpha})\cup\\{y\in(a,b),|y-x_{0}|>r_{k}^{\alpha}\\}=I_{k,1}^{\alpha}\cup
I_{k,2}^{\alpha}$.

We start by providing an $L^{p}$ bound of
${\left\|{\Psi_{R,k}^{\delta}}\right\|}_{L^{p}(I^{\alpha}_{k,1},\omega)}$. To
do so, we proceed in the way to reduce the $L^{p}$ inequality (19) to certain
$(L^{p},L^{2})$ inequality using the last lemma.

Using Parseval formula and the fact that
$\mbox{supp}\left(\phi_{R,k}^{\delta}\right)\subseteq(R_{k,1},R_{k,2})$, where
$R_{k,1}=R(1-2^{-k+1})$ and $R_{k,2}=R(1-2^{-k-1}),$ we have

$\displaystyle{\left\|{\Psi_{R,k}^{\delta}f}\right\|}^{2}_{L^{2}(I,w)}$
$\displaystyle=$
$\displaystyle{\left\|{\sum_{n=0}^{\infty}\phi_{R,k}^{\delta}(\lambda_{n})a_{n}(f)\varphi_{n}}\right\|}^{2}_{L^{2}(I,w)}$
$\displaystyle=$ $\displaystyle{\left\|{\sum_{R_{k,1}\leq\lambda_{n}\leq
R_{k,2}}\phi_{R,k}^{\delta}(\lambda_{n})a_{n}(f)\varphi_{n}}\right\|}^{2}_{L^{2}(I,w)},$

Using the previous lemma with $m=R_{k,1}$, $M=R_{k,2}$ and the fact that
$\displaystyle\sup_{t\in\mathbb{R}}|\phi_{R,k}^{\delta}(t)|\leq
C2^{-k\delta}$, one gets

$\displaystyle{\left\|{\Psi_{R,k}^{\delta}f}\right\|}^{2}_{L^{2}(I,w)}$
$\displaystyle\leq$ $\displaystyle
C2^{-2k\delta}{\left\|{\sum_{R_{k,1}\leq\lambda_{n}\leq
R_{k,2}}a_{n}(f)\varphi_{n}}\right\|}^{2}_{L^{2}(I,w)}$ $\displaystyle\leq$
$\displaystyle
C2^{-2k\delta}R^{(2\frac{\gamma(p^{\prime})}{\varepsilon})}\left(\frac{3R}{2^{k+1}}\right){\left\|{f}\right\|}^{2}_{L^{p}(I,w)}.$

Hence, we have

${\left\|{\Psi_{R,k}^{\delta}f}\right\|}_{L^{2}(I,w)}\leq
C2^{-k(\delta+\frac{1}{2})}R^{(\frac{\gamma(p^{\prime})}{\varepsilon})+\frac{1}{2})}{\left\|{f}\right\|}_{L^{p}(I,w)}.$
(20)

By combining Hölder inequality and (20), we obtain

$\displaystyle{\left\|{\Psi_{R,k}^{\delta}f}\right\|}_{L^{p}(I_{k,1}^{\alpha},w)}$
$\displaystyle\leq$
$\displaystyle(\mu(I_{k,1}))^{\frac{1}{p}-\frac{1}{2}}{\left\|{\Psi_{R,k}^{\delta}f}\right\|}_{L^{2}(I_{k,1}^{\alpha},w)}$
(21) $\displaystyle\leq$
$\displaystyle(2r_{k}^{\alpha})^{\frac{1}{p}-\frac{1}{2}}{\left\|{\Psi_{R,k}^{\delta}f}\right\|}_{L^{2}(I,w)}$
$\displaystyle\leq$ $\displaystyle
C2^{-k(\delta-\frac{\gamma(p^{\prime})}{\varepsilon})}{\left\|{f}\right\|}_{L^{p}(I,w)}.$

Let $\displaystyle
s_{R,k}^{\delta}(u,v)=\sum_{n=0}^{\infty}\phi_{R,k}^{\delta}(\lambda_{n})\varphi_{n}(x)\varphi_{n}(y)$
be the kernel of $\Psi_{R,k}^{\delta}$. We just have to find an estimate of
$||\Psi_{R,k}^{\delta}f||_{L^{p}(I_{k,2}^{\alpha},w)}$, so we will use the
Schur test with the symmetric property of $s_{R,k}^{\delta},$ then it suffices
to prove the following inequality

$\sup_{u\in
I_{k,2}^{\alpha}}{\left\|{s_{R,k}^{\delta}(u,.)}\right\|}_{L^{1}(I_{k,2}^{\alpha})}\leq
C2^{-k\varepsilon}$

for some $\varepsilon>0$ and $C>0$ depending only on $p.$
We consider
$g_{R,k}^{\delta}(\lambda)=\left(1-\frac{\lambda^{2}}{R}\right)_{+}^{\delta}e^{\lambda^{2}/R}\phi(2^{k}(1-\frac{\lambda^{2}}{R}))$
satisfying the following properties, see [8]

  1. 1.

For every non-negative integer $i$ there exists a constant $C_{i}$ such that
for all $s>0$

$\int_{|t|\geq s}|\hat{g}_{R,k}^{\delta}(t)|dt\leq
C_{i}s^{-i}R^{-i/2}2^{(i-\delta)k}$ (22)

  2. 2.

${\left\|{g_{R,k}^{\delta}(\sqrt{\mathcal{L}})}\right\|}_{(L^{2},L^{2})}\leq
C2^{-k\delta}.$ (23)

For our purpose, we will consider such a positive self-adjoint operator
$\mathcal{L}$ on $L^{2}(\mathbb{R})$ such that the semigroup
$e^{-t\mathcal{L}}$, generated by $-\mathcal{L}$, has the kernel $p_{t}(x,y)$
obeying the Gaussian upper bound

$|p_{t}(u,v)|\leq\frac{C}{\sqrt{t}}\exp{\left(-\frac{|u-v|^{2}}{Ct}\right)}.$
(24)

for a constant $C>0$. (see [14])
For all $u\in\mathbb{R}$ and $t>0$, one gets the following estimate

${\left\|{p_{t}(u,.)}\right\|}_{L^{2}(\mathbb{R})}\leq C.$ (25)

On the other hand, there exists $i_{0}\in\mathbb{N}$ such that
$2^{i_{o}-1}<R^{\mu(p)}<2^{i_{0}}$ and we can see that

$I_{k,2}^{\alpha}\subseteq\displaystyle\cup_{\mu(p)k-i_{0}\leq j\leq 0}D_{j}$

where $D_{j}=\\{y,2^{j}r\leq|y-x_{0}|<2^{j+1}r\\}.$ Since, $\mathcal{L}$ is a
positive self-adjoint operator, then it’s clear that

$\phi_{R,k}^{\delta}(\mathcal{L})=g_{R,k}^{\delta}(\sqrt{\mathcal{L}})\exp{\left(-\mathcal{L}/R\right)}.$
(26)

Hence one gets

$\displaystyle s_{R,k}^{\delta}(u,v)$ $\displaystyle=$ $\displaystyle
g_{R,k}^{\delta}(\sqrt{\mathcal{L}})\left(p_{1/R}(u,.)\right)(v)$
$\displaystyle=$ $\displaystyle
g_{R,k}^{\delta}(\sqrt{\mathcal{L}})\left(p_{1/R}(u,.)\chi_{\\{w,|x_{0}-w|<2^{j-1}r\\}}\right)(v)+g_{R,k}^{\delta}(\sqrt{\mathcal{L}})\left(p_{1/R}(u,.)\chi_{\\{w,|x_{0}-w|\geq
2^{j-1}r\\}}\right)(v)$ $\displaystyle=$ $\displaystyle
s_{R,k}^{\delta,1}(u,v)+s_{R,k}^{\delta,2}(u,v).$

Using the fact that $g_{R,k}^{\delta}$ is an even function, with the inversion
formula, we have

$g_{R,k}^{\delta}(\sqrt{\lambda})=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\hat{g}_{R,k}^{\delta}(t)\cos{(t\sqrt{\lambda})}dt.$

Hence, we obtain

$\displaystyle s_{R,k}^{\delta,1}(u,v)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\hat{g}_{R,k}^{\delta}(t)\cos{(t\sqrt{\mathcal{L}})}\left(p_{1/R}(u,.)\chi_{\\{w,|x_{0}-w|<2^{j-1}r\\}}\right)(v)dt.$

Moreover, the operator $\cos{(t\sqrt{\mathcal{L}})}$ is bounded in $L^{2}$
with support kernel $\mathcal{K}_{t}$ satisfying, see [14, 28]

$\mbox{Supp}\left(\mathcal{K}_{t}\right)=\\{(u,v)\in\mathbb{R}^{2},|u-v|\leq
c_{0}|t|\\}$

From (23), (25) and the previous analysis, one gets

$\displaystyle{\left\|{s_{R,k}^{\delta,1}(u,.)}\right\|}_{L^{1}(D_{j})}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}{\left\|{\int_{\mathbb{R}}\hat{g}_{R,k}^{\delta}(t)\cos{(t\sqrt{\mathcal{L}})}\left(p_{1/R}(u,.)\chi_{\\{w,|x_{0}-w|<2^{j-1}r\\}}\right)dt}\right\|}_{L^{1}(D_{j})}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}{\left\|{\int_{|t|>\frac{2^{j-1}r}{c_{0}}}\hat{g}_{R,k}^{\delta}(t)\cos{(t\sqrt{\mathcal{L}})}\left(p_{1/R}(u,.)\chi_{\\{w,|x_{0}-w|<2^{j-1}r\\}}\right)dt}\right\|}_{L^{1}(D_{j})}$
$\displaystyle\leq$
$\displaystyle\frac{\mu^{1/2}(D_{j})}{\sqrt{2\pi}}\int_{|t|>\frac{2^{j-1}r}{c_{0}}}|\hat{g}_{R,k}^{\delta}(t)|{\left\|{p_{1/R}(u,.)}\right\|}_{L^{2}(D_{j})}dt$
$\displaystyle\leq$
$\displaystyle\frac{C}{\sqrt{2\pi}}2^{\frac{j+1}{2}}\int_{|t|>\frac{2^{j-1}r}{c_{0}}}|\hat{g}_{R,k}^{\delta}(t)|dt$

Let $i>\frac{\mu+\frac{1}{2}}{2(\mu+1-\frac{1}{p})}$ where
$\mu=\frac{\gamma(p^{\prime})}{\varepsilon}>0.$ Then by (22), there exists a
constant $C_{i}>0$ such that

$\displaystyle{\left\|{s_{R,k}^{\delta,1}(u,.)}\right\|}_{L^{1}(D_{j})}$
$\displaystyle\leq$
$\displaystyle\frac{C_{i}}{\sqrt{2\pi}}2^{j/2}(\frac{2^{j}}{c_{0}})^{-i}R^{-i/2}2^{(i-\delta)k}$
$\displaystyle\leq$
$\displaystyle\frac{C_{i}}{\sqrt{2\pi}}c_{0}^{i}2^{(i-\delta)k}2^{j(1/2-i)}.$

Then, we obtain

$\displaystyle{\left\|{s_{R,k}^{\delta,1}(u,.)}\right\|}_{L^{1}(I_{k,2})}$
$\displaystyle\leq$ $\displaystyle\sum_{\mu(p)k-i_{0}\leq j\leq
0}{\left\|{s_{R,k}^{\delta,1}(u,.)}\right\|}_{L^{1}(D_{j})}$
$\displaystyle\leq$
$\displaystyle\frac{C_{i}}{\sqrt{2\pi}}c_{0}^{i}2^{(i-\delta)k}\sum_{\mu(p)k-i_{0}\leq
j\leq 0}2^{j(1/2-i)}$ $\displaystyle\leq$
$\displaystyle\frac{C_{i}}{\sqrt{2\pi}}c_{0}^{i}2^{(i-\delta)k}2^{(i-1/2)(i_{0}-\mu(p)k+1)}$
$\displaystyle\leq$ $\displaystyle C^{\prime}_{i}2^{-k\varepsilon_{1}}.$

From our assumption on $i,$
$\varepsilon_{1}=\delta-i+(i-1/2)(\frac{\mu+\frac{1}{2}}{(\frac{1}{p}-\frac{1}{2})}))>0.$
Then, to have an estimate of the kernel $s_{R,k}^{\delta,1}$ on
$L^{1}(I_{k,2}^{\delta})$, it suffices to find an estimate of the kernel
$s_{R,k}^{\delta,2}$ on $L^{1}(I_{k,2}^{\delta})$.
From (23), (24) and using the fact that $R\leq R^{\mu(p)}$, one gets the
following inequality

$\displaystyle{\left\|{s_{R,k}^{\delta,2}(u,.)}\right\|}_{L^{1}(D_{j})}$
$\displaystyle=$
$\displaystyle\int_{D_{j}}|g_{R,k}^{\delta}(\sqrt{\mathcal{L}})\left(p_{1/R}(u,.)\chi_{\\{w,|w-x_{0}|>2^{j-1}r\\}}\right)(v)|dv$
$\displaystyle\leq$
$\displaystyle{\left\|{g_{R,k}^{\delta}(\sqrt{\mathcal{L}})}\right\|}_{(L^{2},L^{2})}{\left\|{p_{1/R}(u,.)\chi_{\\{w,|w-x_{0}|>2^{j-1}r\\}}}\right\|}_{L^{2}(D_{j})}$
$\displaystyle\leq$ $\displaystyle
C2^{-k\delta}\sqrt{R}e^{(-CR2^{2j-2})}\left(\mu(D_{j})\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle
C2^{-k\delta}2^{\frac{i_{0}+j}{2}}e^{-C2^{2(i_{0}+j)}}.$

Hence, we conclude that

$\displaystyle{\left\|{s_{R,k}^{\delta,2}(u,.)}\right\|}_{L^{1}(I_{k,2})}$
$\displaystyle\leq$ $\displaystyle\sum_{\mu(p)k-i_{0}\leq j\leq
0}{\left\|{s_{R,k}^{\delta,2}(u,.)}\right\|}_{L^{1}(D_{j})}$
$\displaystyle\leq$ $\displaystyle
C2^{-k\delta}\sum_{i=i_{o}+j\geq\mu(p)k}2^{\frac{i}{2}}e^{-C2^{2i}}$
$\displaystyle\leq$ $\displaystyle C^{\prime}2^{-k\delta}.$

∎

######  Proposition 2.

Let $1\leq p\leq 2$ and
$\delta>\delta(p)=\frac{\gamma(p^{\prime})}{\varepsilon}$, then for all $f\in
L^{p}(I,w)$, we have

${\left\|{\psi_{R,0}^{\delta}f}\right\|}_{L^{p}(I,w)}\leq
C{\left\|{f}\right\|}_{L^{p}(I,w)}.$ (27)

where $C$ is a constant independent of $f$ and $R.$

###### Proof.

It suffices to use the same techniques as those used in the previous proof to
get an estimate of ${\left\|{\psi_{R,0}^{\delta}f}\right\|}_{L^{p}(I_{1},w)}$
and ${\left\|{\psi_{R,0}^{\delta}f}\right\|}_{L^{p}(I_{2},w)}$ for all $f\in
L^{p}(I,w),$ where $I=(a,b)=I_{1}\cup I_{2}$ with
$I_{1}=(x_{0}-r^{\alpha}_{0},x_{0}+r^{\alpha}_{0})$ and
$I_{2}=\\{y,|y-x_{0}|>r^{\alpha}_{0}\\}$ where
$r^{\alpha}_{0}=\frac{r}{R^{\mu(p)}}.$ ∎

To conclude the theorem’s proof it suffices to find a uniform bound of
$\mathcal{R}_{R}^{\delta}$.

######  Proposition 3.

Let $1\leq p\leq 2$ and
$\delta>\delta(p)=\frac{\gamma(p^{\prime})}{\varepsilon}$, then for all $f\in
L^{p}(I,w)$, we have

${\left\|{\mathcal{R}_{R}^{\delta}f}\right\|}_{L^{p}(I,w)}\leq
C{\left\|{f}\right\|}_{L^{p}(I,w)}.$ (28)

where $C$ depends only on $p$.

###### Proof.

From Holder’s inequality and the previous lemma, we have

$\displaystyle{\left\|{\mathcal{R}_{R}^{\delta}f}\right\|}^{2}_{L^{p}(I,w)}$
$\displaystyle\leq$ $\displaystyle
2^{2(\frac{1}{p}-\frac{1}{2})}{\left\|{\mathcal{R}_{R}^{\delta}f}\right\|}^{2}_{L^{2}(I,w)}$
$\displaystyle\leq$ $\displaystyle
2^{2(\frac{1}{p}-\frac{1}{2})}\sum_{k=K_{R}+1}^{\infty}\sum_{n=0}^{\infty}{\left\|{\phi_{R,k}^{\delta}(\lambda_{n})a_{n}(f)\varphi_{n}}\right\|}_{L^{2}(I,w)}$
$\displaystyle\leq$ $\displaystyle
C2^{2(\frac{1}{p}-\frac{1}{2})}\sum_{k=K_{R}+1}^{\infty}2^{-2k\delta}\sum_{R_{k,1}\leq\lambda_{n}\leq
R_{k,2}}{\left\|{a_{n}(f)\varphi_{n}}\right\|}^{2}_{L^{2}(I,w)}$
$\displaystyle\leq$ $\displaystyle
C2^{2(\frac{1}{p}-\frac{1}{2})}\sum_{k=K_{R}+1}^{\infty}2^{-2k(\delta+\frac{1}{2})}R^{2(\frac{1}{2}+\frac{\gamma(p^{\prime})}{\varepsilon})}{\left\|{f}\right\|}^{2}_{L^{p}(I,\omega)}$
$\displaystyle\leq$ $\displaystyle
C2^{2(\frac{1}{p}-\frac{1}{2})}2^{-2(\delta+\frac{1}{2})\big{(}\left[\frac{\log(R)}{\log(2)}\right]+1\big{)}}R^{2(\frac{1}{2}+\frac{\gamma(p^{\prime})}{\varepsilon})}{\left\|{f}\right\|}^{2}_{L^{p}(I,\omega)}$
$\displaystyle\leq$ $\displaystyle
C2^{2(\frac{1}{p}-\frac{1}{2})}R^{-2(\delta-(\frac{\gamma(p^{\prime})}{\varepsilon}))}{\left\|{f}\right\|}^{2}_{L^{p}(I,\omega)}$

Finally we obtain

$\displaystyle{\left\|{\mathcal{R}_{R}^{\delta}f}\right\|}_{L^{p}(I,w_{\alpha,\beta})}$
$\displaystyle\leq$ $\displaystyle
C2^{(\frac{1}{p}-\frac{1}{2})}R^{-(\delta-(\frac{\gamma(p^{\prime})}{\varepsilon}))}{\left\|{f}\right\|}_{L^{p}(I,\omega_{\alpha,\beta})}$
$\displaystyle\leq$ $\displaystyle
C(p){\left\|{f}\right\|}_{L^{p}(I,\omega_{\alpha,\beta})}.$

∎

∎

######  Corollary 1.

Under the notation and conditions of the previous Theorem, we have for all
$f\in L^{p}(I,w)$

$\Psi_{R}^{\delta}f\to f~{}~{}\mbox{as}~{}~{}R\to\infty.$ (29)

###### Proof.

Step1:  We prove that, for every $f\in\mathcal{C}^{\infty}(I,\mathbb{R})$,
$\Psi^{\delta}_{R}f\to f$ in $L^{p}(I,\omega)$. Note that

$\displaystyle\Big{|}\Big{(}1-\frac{\lambda_{n}}{R}\Big{)}^{\delta}_{+}{\left\langle{f,\varphi_{n}}\right\rangle}_{L^{2}(I,\omega)}\Big{|}$
$\displaystyle\leq$
$\displaystyle\Big{|}{\left\langle{f,\varphi_{n}}\right\rangle}_{L^{2}(I,\omega)}\Big{|}=\frac{1}{\lambda_{n}}\Big{|}{\left\langle{f,\mathcal{L}\varphi_{n}}\right\rangle}_{L^{2}(I,\omega)}\Big{|}$
(30) $\displaystyle=$
$\displaystyle\frac{1}{\lambda_{n}}\Big{|}{\left\langle{\mathcal{L}f,\varphi_{n}}\right\rangle}_{L^{2}(I,\omega)}\Big{|}=\cdots=\frac{1}{\lambda_{n}^{k}}\Big{|}{\left\langle{\mathcal{L}^{k}.f,\varphi_{n}}\right\rangle}_{L^{2}(I,\omega)}\Big{|}$
$\displaystyle\leq$ $\displaystyle
n^{-k\varepsilon}{\left\|{\mathcal{L}^{k}.f}\right\|}_{L^{2}(I,\omega)}.$

Since ${\left\|{\varphi_{n}}\right\|}_{L^{2}(I,\omega)}\leq n^{\gamma(p)}$, it
suffices to take $k$ big enough to have $\gamma(p)-k\varepsilon<-1$ and obtain
the convergence of the series in $L^{p}(I,\omega)$.
Since
$\displaystyle{\left\|{\Psi^{\delta}_{R}.f-f}\right\|}_{2}^{2}=\sum_{n=0}^{\infty}\Big{(}(1-\frac{\lambda_{n}}{R})^{\delta}_{+}-1\Big{)}^{2}|a_{n}(f)|^{2}\to
0$ as $R\to\infty$,then the result remains true for $1\leq p<\infty$.
Step2: For all $\varepsilon>0$. By density of
$\mathcal{C}^{\infty}_{0}(I,\mathbb{R})$ in $L^{p}(I,\omega)$, there exists
$g\in\mathcal{C}^{\infty}_{0}(I,\mathbb{R})$ such that
${\left\|{f-g}\right\|}_{L^{p}(I,\omega)}<\varepsilon$ and there exists $R>0$
such that
${\left\|{\Psi^{\delta}_{R}.f-\Psi^{\delta}_{R}.g}\right\|}_{L^{p}(I,\omega)}<\varepsilon$.
By writing,

${\left\|{\Psi^{\delta}_{R}.f-f}\right\|}_{L^{p}(I,\omega)}\leq{\left\|{\Psi^{\delta}_{R}.f-\Psi^{\delta}_{R}.g}\right\|}_{L^{p}(I,\omega)}+{\left\|{\Psi^{\delta}_{R}.g-g}\right\|}_{L^{p}(I,\omega)}+{\left\|{f-g}\right\|}_{L^{p}(I,\omega)},$

one gets the desired result. ∎

To conclude for the proof of sufficient conditions of both theorems 1 and 2,
it suffices to verify that the two considered bases satisfy conditions (A) and
(B).We will prove this result only for the case of GPSWFs. The other case is
almost identical.
We first recall that from (13), the GPSWFs are the eigenfunctions of the
Sturm-Liouville operator $\mathcal{L}_{c}^{(\alpha)}.$ Also, note that the
$(n+1)-$th eigenvalue $\chi_{n,\alpha}(c)$ of $\mathcal{L}_{c}^{(\alpha)}$
satisfies the following classical inequalities,

$n^{2}\leq n(n+2\alpha+1)\leq\chi_{n,\alpha}(c)\leq
n(n+2\alpha+1)+c^{2},\quad\forall n\geq 0.$

Moreover, for every $0\leq m<M$ such that $M-m>1$, we have

$\displaystyle\sum_{\chi_{n,\alpha}(c)\in(m,M)}1$ $\displaystyle\leq$
$\displaystyle\sum_{n(n+2\alpha+1)\in(\max(0,m-c^{2}),M)}1$
$\displaystyle\leq$
$\displaystyle\sum_{(n+\alpha+1/2)^{2}-(\alpha+1/2)^{2}\in(\max(0,m-c^{2}),M)}1$
$\displaystyle\leq$
$\displaystyle\sum_{n\in\left((\max(0,m-c^{2})+(\alpha+1/2)^{2})^{\frac{1}{2}}-(1/2+\alpha),(M+(\alpha+1/2)^{2})^{\frac{1}{2}}-(1/2+\alpha)\right)}1$
$\displaystyle\leq$ $\displaystyle C(M-m).$

It follows that condition (B) is satisfied.
Form [6] Lemma $2.6$, one can conclude that condition (A) is satisfied for
weighted prolate spheroidal wave functions for $1<p<\infty$. Moreover, it has
been shown in [18] that ${\left\|{\psi^{(\alpha)}_{n,c}}\right\|}_{\infty}\leq
C\Big{(}\chi_{n,\alpha}(c)\Big{)}^{\frac{\alpha+1}{2}}.$ Then, by using (14),
we obtain ${\left\|{\psi^{(\alpha)}_{n,c}}\right\|}_{1}\leq
C\Big{(}\chi_{n,\alpha}(c)\Big{)}^{\frac{\alpha+1}{2}}\leq Cn^{\alpha+1}.$

######  Remark 3.

The uniform norm of the CPSWFs has been given in [5].

##  5 Proof of necessary condition

The transferring theorem from the uniform boundedness of $\Psi_{R}^{\delta}$
to the uniform boundedness of the Hankel multiplier transform operator
$\mathcal{M}_{\alpha}$ defined by
$\mathcal{M}_{\alpha}(f)=\mathcal{H}_{\alpha}\left(\phi(.)\mathcal{H}_{\alpha}(f)\right)$
can be used to derive necessary condition. Note here that $\phi$ is a bounded
function on $\mathbb{R}$, continuous except on a set of Lebesgue measure zero
and $\mathcal{H}_{\alpha}$ is the modified Hankel operator defined by

$\mathcal{H}_{\alpha}(f)(x)=\int_{0}^{\infty}\frac{J_{\alpha}(xy)}{(xy)^{\alpha}}f(y)y^{2\alpha+1}dy.$

From [12] and the transferring theorem, the uniform boundedness of
$\Psi_{R}^{\delta}$ holds true if and only if
$\delta>\max\\{2(\alpha+1)|\frac{1}{p}-\frac{1}{2}|-\frac{1}{2},0\\}.$ It’s
easy to check that
$\max\\{2(\alpha+1)|\frac{1}{p}-\frac{1}{2}|-\frac{1}{2},0\\}\geq\max\\{\frac{\gamma_{\alpha}(p^{\prime})}{2},0\\}$
for every $p\not=2-\frac{1}{\alpha+3/2}$, then one gets our necessary
condition. To be more precise, let’s study each transferring theorem
separately.

###  5.1 GPSWFs’s case

Let’s recall that the family of weighted prolate spheroidal wave functions
$\\{\psi_{n,c}^{(\alpha)}(\cos\theta)\\}_{n}$ form an orthonormal system on
$(0,\pi)$ with respect to the measure $(\sin\theta)^{2\alpha+1}d\theta$.
For a function $f(\theta)$ integrable on $(0,\pi)$ with respect to the measure
defined above, we have formally

$f(\theta)=\sum_{n=0}^{\infty}a_{n}(f)\psi_{n,c}^{(\alpha)}(\cos\theta)\qquad
a_{n}(f)=\int_{0}^{\pi}f(\theta)\psi_{n,c}^{(\alpha)}(\cos\theta)(\sin\theta)^{2\alpha+1}d\theta$

For $p\geq 1$ and a function $f$ on $(0,\pi)$ we define a norm

${\left\|{f}\right\|}_{p}=\Bigg{(}\int_{0}^{\pi}|f(\theta)|^{p}(\sin\theta)^{2\alpha+1}\Bigg{)}^{1/p}.$

Before stating an adequate transferring theorem, let’s define a GPSWFs-
multiplier.

######  Definition 1.

Let $\lambda>0$ be a sufficiently large real, the bounded sequence
$\\{\phi(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})\\}_{n}$ is called a
Weighted prolate multiplier if there exist a constant $C>0$ such that for
every $f\in L^{p}(I,\omega_{\alpha})$, we have

${\left\|{\sum_{n=0}^{\infty}\phi(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})a_{n}(f)\psi_{n,c}^{(\alpha)}}\right\|}_{L^{p}(I,\omega_{\alpha})}\leq
C{\left\|{f}\right\|}_{L^{p}(I,\omega_{\alpha})}.$

The smallest constant $C$ verifying this last inequality is written
${\left\|{\phi(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})}\right\|}_{p}$. In
the same context, the function $\phi$ is called an $\L^{p}$-Hankel transform
multiplier if
$\mathcal{M}_{\alpha}(f)=\mathcal{H}_{\alpha}(\phi(.)\mathcal{H}_{\alpha}(f))$
is uniformly bounded on
$L^{p}\left((0,\infty),\theta^{2\alpha+1}d\theta\right)$.

######  Theorem 4 (Transferring theorem).

Let $1<p<\infty$, $0\leq\alpha<3/2$ and $\phi$ be a bounded function on
$(0,\infty)$ continuous except on a set of Lebesgue measure zero such that
$\\{\phi(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})\\}_{n}$ is a Weighted
prolate multiplier for all large $\lambda>0$ and
$\displaystyle\liminf_{\lambda\to\infty}{\left\|{\phi(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})}\right\|}_{p}$
is finite then $\phi$ is an $L^{p}$-Hankel transform multiplier and we have

${\left\|{\mathcal{M}_{\alpha}}\right\|}_{p}\leq\liminf_{\lambda\to\infty}{\left\|{\phi\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\right)}\right\|}_{p}.$

###### Proof.

Let $g$ be an infinitely differentiable function with compact support in
$[0,M]$ and put $g_{\lambda}(\theta)=g(\lambda\theta)$. Here $\lambda$ is a
positive real so that $supp(g_{\lambda})\subset[0,\pi]$.
Recall that we have by assumption

${\left\|{\sum_{n=0}^{\infty}\phi(\chi^{1/2}_{n,\alpha}(c)/\lambda)a_{n}(g)\psi_{n,c}^{(\alpha)}(\cos(.))}\right\|}_{p}\leq{\left\|{\phi(\chi^{1/2}_{n,\alpha}(c)/\lambda)}\right\|}_{p}{\left\|{g}\right\|}_{p}.$
(31)

Via a simple change of variable, one can write

$\lim_{\lambda\to\infty}\lambda^{2\alpha+2}{\left\|{g_{\lambda}}\right\|}_{p}^{p}=\lim_{\lambda\to\infty}\int_{0}^{M}|g(\tau)|^{p}\Big{(}\lambda\sin(\tau/\lambda)\Big{)}^{2\alpha+1}d\tau=\int_{0}^{\infty}|g(\tau)|^{p}\tau^{2\alpha+1}d\tau.$

By using (31) together with Fatou’s lemma, one gets

$\displaystyle\displaystyle\int_{0}^{\infty}$
$\displaystyle\liminf_{\lambda\to\infty}\Big{|}\chi_{(0,\pi\lambda)}(\tau)\sum_{n=0}^{\infty}\phi(\chi^{1/2}_{n,\alpha}(c)/\lambda)a_{n}(g_{\lambda})\psi_{n,c}^{(\alpha)}(\cos\tau/\lambda)\Big{|}^{p}\tau^{2\alpha+1}d\tau$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\liminf_{\lambda\to\infty}\Big{|}\chi_{(0,\pi\lambda)}(\tau)\sum_{n=0}^{\infty}\phi(\chi^{1/2}_{n,\alpha}(c)/\lambda)a_{n}(g_{\lambda})\psi_{n,c}^{(\alpha)}(\cos\tau/\lambda)\Big{|}^{p}\lambda^{2\alpha+1}\sin(\tau/\lambda)^{2\alpha+1}d\tau$
$\displaystyle\leq$
$\displaystyle\liminf_{\lambda\to\infty}\lambda^{2\alpha+1}\int_{0}^{\infty}\Big{|}\chi_{(0,\pi\lambda)}(\tau)\sum_{n=0}^{\infty}\phi(\chi^{1/2}_{n,\alpha}(c)/\lambda)a_{n}(g_{\lambda})\psi_{n,c}^{(\alpha)}(\cos\tau/\lambda)\Big{|}^{p}\sin(\tau/\lambda)^{2\alpha+1}d\tau$
$\displaystyle\leq$
$\displaystyle\liminf_{\lambda\to\infty}\lambda^{2\alpha+2}{\left\|{\phi(\chi^{1/2}_{n,\alpha}(c)/\lambda)}\right\|}_{p}{\left\|{g_{\lambda}}\right\|}^{p}_{p}=\liminf_{\lambda\to\infty}{\left\|{\phi(\chi^{1/2}_{n,\alpha}(c)/\lambda)}\right\|}_{p}\Bigg{[}\int_{0}^{\infty}|g(\tau)|^{p}\tau^{2\alpha+1}d\tau\Bigg{]}.$

Then there exists a sequence
$\lambda_{1}<\lambda_{2}<\cdots<\lambda_{p}\to\infty$ that
$G(\tau,\lambda)=\displaystyle\chi_{(0,\pi\lambda)}(\tau)\sum_{n=0}^{\infty}\phi(\chi^{1/2}_{n,\alpha}(c)/\lambda)a_{n}(g_{\lambda})\psi_{n,c}^{(\alpha)}(\cos\frac{\tau}{\lambda})$
converges weakly to a function $G(\tau)$. Furthermore, $G$ satisfies

$\Bigg{[}\int_{0}^{\infty}|G(\tau)|^{p}\tau^{2\alpha+1}d\tau\Bigg{]}^{1/p}\leq\liminf_{\lambda\to\infty}{\left\|{\phi(\chi^{1/2}_{n,\alpha}(c)/\lambda)}\right\|}_{p}\Bigg{[}\int_{0}^{\infty}|g(\tau)|^{p}\tau^{2\alpha+1}d\tau\Bigg{]}^{1/p}.$

Let us now prove that $G=\mathcal{H}_{\alpha}(\phi.\mathcal{H}_{\alpha}(g))$.
Let

$G(\tau,\lambda)=\chi_{(0,\pi\lambda)}(\tau)\Big{[}\sum_{n=0}^{N[\lambda]}+\sum_{N[\lambda]+1}^{\infty}\Big{]}\phi(\chi^{1/2}_{n,\alpha}(c)/\lambda)a_{n}(g_{\lambda})\psi_{n,c}^{(\alpha)}(\cos\tau/\lambda)=G^{N}(\tau,\lambda)+H^{N}(\tau,\lambda)$

We start by giving the following lemma that will be proved later,

######  Lemma 2.

We have

$\int_{0}^{\infty}\Big{|}H^{N}(\tau,\lambda)\Big{|}^{2}\tau^{2\alpha+1}d\tau=O(\frac{1}{N})\mbox{
uniformly in }\lambda$

Therefore, by the diagonal argument, there exists a subsequence also noted
$\\{\lambda_{j}\\}$ for a sake of clarity, such that $H^{N}(\tau,\lambda_{j})$
converges weakly to a function $H^{N}(\tau)$ and

$\displaystyle\int_{0}^{\pi}\Big{|}H^{N}(\tau)\Big{|}^{p}\tau^{2\alpha+1}d\tau=O(\frac{1}{N^{2}}).$

Then, there exists a subsequence $H^{N_{j}}$ denoted for the same reason
$H^{N}$ that converges to zero a.e .
Since $G^{N}(\tau,\lambda)=G(\tau,\lambda)-H^{N}(\tau,\lambda)$,
$G^{N}(\tau,\lambda)$ converge weakly to a limit $G^{N}(\tau)$ and
$G(\tau)=G^{N}(\tau)+H^{N}(\tau)$. Thus $G^{N}(\tau)$ converges to $G(\tau)$
almost everywhere. On the other hand, we will prove the following lemma

######  Lemma 3.

We have

$\lim_{\lambda\to\infty}G^{N}(\tau,\lambda)=\int_{0}^{N}\phi(v)\mathcal{H}_{\alpha}.g(v)\frac{J_{\alpha}(v\tau)}{(v\tau)^{\alpha}}v^{2\alpha+1}dv,$

which implies that

$G(\tau)=\int_{0}^{\infty}\phi(v)\mathcal{H}_{\alpha}.g(v)\frac{J_{\alpha}(v\tau)}{(v\tau)^{\alpha}}v^{2\alpha+1}dv,$

and achieves our proof. ∎

###### Proof of Lemma 2.

We have

$\displaystyle\int_{0}^{M}|H^{N}(\tau,\lambda)|^{2}\Big{(}\lambda\sin\frac{\tau}{\lambda}\Big{)}^{2\alpha+1}d\tau$
$\displaystyle=$
$\displaystyle\lambda^{2\alpha+2}\int_{0}^{\pi}|H^{N}(\lambda\tau,\lambda)|^{2}(\sin\tau)^{2\alpha+1}d\tau$
(32) $\displaystyle=$
$\displaystyle\lambda^{2\alpha+2}\sum_{N[\lambda]+1}^{\infty}|\phi(\frac{n}{\lambda})|^{2}|a_{n}(g_{\lambda})|^{2}.$

Recall that in [19], authors have given the following uniform approximation of
GPSWFs in term of Jacobi polynomials for $0\leq\alpha<3/2$,

$\psi_{n,c}^{(\alpha)}(\cos\theta)=A_{n}\widetilde{P}_{n}^{(\alpha)}(\cos\theta)+R_{n,c}^{(\alpha)}(\cos\theta)\qquad{\left\|{R_{n,c}}\right\|}^{(\alpha)}_{\infty}\leq
C_{\alpha,c}\frac{1}{2n+2\alpha+1}.$ (33)

We also know that (see for example [29])

$n(\sin\theta)^{2\alpha+1}\widetilde{P}_{n}^{(\alpha,\alpha)}(\cos\theta)=2\frac{h^{\alpha+1}_{n-1}}{h^{(\alpha)}_{n}}\frac{d}{d\theta}\Big{[}(\sin\theta)^{2\alpha+2}\widetilde{P}_{n-1}^{(\alpha+1,\alpha+1)}(\cos\theta)\Big{]}$
(34)

By combining (33) and (34), one gets

$(\sin\theta)^{2\alpha+1}\psi_{n,c}^{(\alpha)}(\cos\theta)=\frac{2}{n}\frac{h^{\alpha+1}_{n-1}}{h^{(\alpha)}_{n}}\frac{d}{d\theta}\Big{[}(\sin\theta)^{2\alpha+2}\widetilde{P}_{n-1}^{(\alpha+1,\alpha+1)}(\cos\theta)\Big{]}+R_{n,c}^{(\alpha)}(\cos\theta).$

Then, integrating by parts one gets

$\displaystyle a_{n}(g_{\lambda})$ $\displaystyle=$
$\displaystyle\frac{C}{n}\int_{0}^{\pi}\frac{g^{\prime}(\lambda\theta)}{\sin\theta}\widetilde{P}_{n-1}^{\alpha+1}(\cos\theta)(\sin\theta)^{2\alpha+3}d\theta+\int_{0}^{\pi}R_{n,c}^{(\alpha)}(\cos\theta)g(\lambda\theta)d\theta$
$\displaystyle=$ $\displaystyle a_{n,1}(g_{\lambda})+a_{n,2}(g_{\lambda})$

Let’s come back to (32). We have by Bessel’s inequality

$\displaystyle\lambda^{2\alpha+2}\sum_{N[\lambda]+1}^{\infty}|\phi(\frac{n}{\lambda})|^{2}|a_{n,1}(g_{\lambda})|^{2}$
$\displaystyle\leq$ $\displaystyle
C\lambda^{2\alpha+2}\Big{[}\frac{\lambda}{N(\lambda-1)}\Big{]}^{2}\sum_{N[\lambda]+1}^{\infty}|\frac{n}{\lambda}a_{n,1}(g_{\lambda})|^{2}$
(35) $\displaystyle\leq$
$\displaystyle\frac{C}{N^{2}}\lambda^{2\alpha+2}\int_{0}^{\pi}\Big{|}\frac{g^{\prime}(\lambda\theta)}{\sin\theta}\Big{|}^{2}(\sin\theta)^{2\alpha+3}d\theta$
$\displaystyle=$
$\displaystyle\frac{C}{N^{2}}\int_{0}^{M}|g^{\prime}(\theta)|^{2}\Big{(}\lambda\sin\frac{\theta}{\lambda}\Big{)}^{2\alpha+1}d\theta$
$\displaystyle=$ $\displaystyle O(\frac{1}{N^{2}})\mbox{ uniformly in
}\lambda.$

On the other hand, using Cauchy-Schwarz’s inequality

$\displaystyle\lambda^{2\alpha+2}\sum_{N[\lambda]+1}^{\infty}|\phi(\frac{n}{\lambda})|^{2}|a_{n,2}(g_{\lambda})|^{2}$
$\displaystyle\leq$ $\displaystyle
C\lambda^{2\alpha+2}\sum_{N[\lambda]+1}^{\infty}{\left\|{R_{n,c}^{\alpha}}\right\|}^{2}_{2}{\left\|{g(\lambda.)}\right\|}_{2}^{2}$
$\displaystyle\leq$ $\displaystyle
C\sum_{N[\lambda]+1}^{\infty}\frac{1}{n^{2}}\int_{0}^{M}|g(\theta)|^{2}\big{(}\lambda\sin\frac{\theta}{\lambda}\big{)}^{2\alpha+1}d\theta$
$\displaystyle=$ $\displaystyle O(\frac{1}{N}).$

Then, one conclude that

$\int_{0}^{M}|H^{N}(\tau,\lambda)|^{2}\tau^{2\alpha+1}d\tau=O(\frac{1}{N})\mbox{
uniformly in }\lambda.$

∎

###### Proof of lemma 3.

We use now the following uniform approximation of GPSWFs in term of Bessel
function (we refer the reader once again to [19])

$\psi_{n,c}^{(\alpha)}(\cos\frac{\tau}{\lambda})=A_{\alpha}(q)\frac{\chi_{n,c}^{1/4}S(\cos\frac{\tau}{\lambda})^{1/2}J_{\alpha}(\chi_{n,c}^{1/2}S(\cos\frac{\tau}{\lambda}))}{(\sin\frac{\tau}{\lambda})^{\alpha+1/2}(1-q\cos^{2}\frac{\tau}{\lambda})^{1/4}}+E_{n,c}(\cos\frac{\tau}{\lambda}),$
(36)

where

$\Big{|}E_{n,c}(\cos\theta)\Big{|}\leq\frac{C.A_{\alpha}(q)}{(1-q)}\frac{(\sin\theta)^{1/2}}{(1-q\cos^{2}\theta)^{1/4}}\qquad\forall\theta\in[0,\pi]\quad\mbox{and}\quad
S(x)=\int_{x}^{1}\sqrt{\frac{1-qt^{2}}{1-t^{2}}}dt.$

Note that it has also been shown in [4] that

$\frac{\sin\theta\sqrt{1-q\cos^{2}\theta}}{S(\cos\theta)}=1+\Big{(}\frac{q}{1-q}+\frac{3}{4}\Big{)}(1-\cos\theta)+o(1-\cos\theta).$

Thus, we can write, for $n\leq N[\lambda]$,and by taking into account that
$\sqrt{x}J_{\alpha}(x)$ is bounded then

$\displaystyle\frac{\psi_{n,c}^{(\alpha)}(\cos\frac{\tau}{\lambda})}{\lambda^{\alpha}}$
$\displaystyle=$ $\displaystyle
n^{1/2}\frac{J_{\alpha}(\frac{n\tau}{\lambda})}{\Big{(}\lambda.\sin\frac{\tau}{\lambda}\Big{)}^{\alpha}}-n^{1/2}\frac{J_{\alpha}(\frac{n\tau}{\lambda})}{\Big{(}\lambda.\sin\frac{\tau}{\lambda}\Big{)}^{\alpha}}\big{(}\frac{q}{1-q}+3/4\big{)}\frac{\tau^{2}}{4\lambda^{\alpha+2}}+O(\frac{1}{n.\lambda^{\alpha+2}})$
(37) $\displaystyle=$ $\displaystyle
n^{1/2}J_{\alpha}(\frac{n\tau}{\lambda})\Big{(}\frac{1}{\tau}\Big{)}^{\alpha}+o(\frac{1}{n}).$

On the other hand,

$\displaystyle\lambda^{\alpha}a_{n}(g_{\lambda})$ $\displaystyle=$
$\displaystyle\lambda^{\alpha-1}\int_{0}^{M}g(\tau)\psi_{n,c}^{(\alpha)}(\cos\frac{\tau}{\lambda})(\sin\frac{\tau}{\lambda})^{2\alpha+1}d\tau$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda^{2}}\Bigg{[}A_{\alpha}(q)n^{1/2}\int_{0}^{\infty}g(\tau)J_{\alpha}(\frac{n\tau}{\lambda})\Big{(}\lambda\sin\frac{\tau}{\lambda}\Big{)}^{\alpha+1}d\tau\Bigg{]}+o(\frac{1}{\lambda^{2}})$
$\displaystyle=$
$\displaystyle\frac{n^{1/2}}{\lambda^{2}}\int_{0}^{\infty}g(\tau)J_{\alpha}(\frac{n\tau}{\lambda})\tau^{\alpha+1}d\tau+o(\frac{1}{\lambda^{2}}).$

Then, by combining the last two estimates, one gets

$G^{N}(\tau,\lambda)=\sum_{n=0}^{N[\lambda]+1}\phi(\frac{n}{\lambda})\mathcal{H}_{\alpha}.g(\frac{n}{\lambda})J_{\alpha}(\frac{n\tau}{\lambda})\frac{1}{\tau^{\alpha}}\frac{n}{\lambda^{2}}+\frac{n}{\lambda^{2}}o(1).$

Therefore, by letting $\lambda\to\infty$, we conclude for the proof of lemma3.

∎

###  5.2 CPSWF’s case

As for the example studied in the previous section, we start by establishing
an adequate transferring theorem for the circular case. To do this, we
introduce a suitable terminology.

######  Definition 2.

Let $\lambda>0$ be a sufficiently large real, a bounded sequence
$\\{m(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})\\}_{n}$ is called to be a
Circular prolate multiplier, if there exists a constant $C>0$ such that for
every $f\in L^{p}(0,1)$, we have

${\left\|{\sum_{n=0}^{\infty}m(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})a_{n}(f)\varphi^{(\alpha)}_{n,c}}\right\|}_{L^{p}(0,1)}\leq
C{\left\|{f}\right\|}_{L^{p}(0,1)}.$

The smallest constant $C$ verifying the last inequality is written
${\left\|{m\Big{(}\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\Big{)}}\right\|}_{p}$.

Here
$\mathcal{M}:=\mathcal{M}_{0}=\mathcal{H}_{0}\Big{(}m(.)\mathcal{H}_{0}(f)\Big{)}$
is the multiplier related to the Hankel transform operator.

######  Theorem 5 (Circular transferring theorem).

Let $1<p<\infty$, $\alpha\geq 1/2$ and $m$ be a bounded function on
$(0,\infty)$ continuous except on a set of Lebesgue measure zero such that
$\\{m(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})\\}_{n}$ is a Circular prolate
multiplier for all large $\lambda>0$ and
$\displaystyle\liminf_{\lambda\to\infty}{\left\|{m(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})}\right\|}_{p}$
is finite then $m$ is an $L^{p}$-Hankel transform multiplier and we have

${\left\|{\mathcal{M}}\right\|}_{p}\leq\liminf_{\lambda\to\infty}{\left\|{m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\right)}\right\|}_{p}.$

###### Proof.

Let $\lambda>0$ and $g\in C^{\infty}_{c}(0,\infty)$ supported in $(0,M)$ such
that $\lambda>\frac{2}{\pi}M$. Let $g_{\lambda}(\tau)=g(\lambda\tau)$ for
every $\tau\in(0,1)$ and $G_{\lambda}=g_{\lambda}\circ\arccos.$

By asymption, we have

${\left\|{\sum_{n=0}^{\infty}m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\right)a_{n}(G_{\lambda})\varphi_{n}}\right\|}_{L^{p}\left(0,1\right)}\leq{\left\|{m(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})}\right\|}_{p}{\left\|{G_{\lambda}}\right\|}_{L^{p}\left(0,1\right)}.$

Then, we get

${\left\|{\chi_{(0,\lambda\frac{\pi}{2})}\sum_{n=0}^{\infty}m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\right)a_{n}\left(G_{\lambda}\right)\varphi_{n}(\cos(\frac{.}{\lambda}))}\right\|}^{p}_{L^{p}((0,\infty),\sin(\frac{.}{\lambda}))}\leq{\left\|{m(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})}\right\|}^{p}_{p}{\left\|{g}\right\|}^{p}_{L^{p}\left((0,\infty),\sin(\frac{.}{\lambda})\right)}$

We denote by

$F_{\lambda}(\theta)=\chi_{(0,\lambda\frac{\pi}{2})}(\theta)\sum_{n=0}^{\infty}m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\right)a_{n}\left(G_{\lambda}\right)\varphi_{n}\left(\cos(\frac{\theta}{\lambda})\right),$

hence we have

${\left\|{F_{\lambda}}\right\|}^{p}_{L^{p}((0,\infty),\sin(\frac{.}{\lambda}))}\leq{\left\|{m(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})}\right\|}^{p}_{p}{\left\|{g}\right\|}^{p}_{L^{p}((0,\infty),\sin(\frac{.}{\lambda}))}.$
(38)

By using (38), Fatou’s Lemma and the fact that
$\displaystyle\lim_{\lambda\to\infty}\lambda\sin(\frac{\theta}{\lambda})=\theta$,
we obtain

${\left\|{\displaystyle\liminf_{\lambda\to\infty}F_{\lambda}}\right\|}^{p}_{L^{p}((0,\infty),\theta
d\theta)}\leq\displaystyle\liminf_{\lambda\to\infty}{\left\|{m(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})}\right\|}^{p}_{p}{\left\|{g}\right\|}^{p}_{L^{p}((0,\infty),\theta
d\theta)}$ (39)

Let
$L=\displaystyle\liminf_{\lambda\to\infty}{\left\|{m(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})}\right\|}_{p}$,
then there exists a sequence of $(\lambda_{j})_{j\in\mathbb{N}}$ such that
$\displaystyle\lim_{j\to\infty}\lambda_{j}=+\infty$ verifying

${\left\|{F_{\lambda_{j}}}\right\|}_{L^{p}((0,\infty),\theta
d\theta)}\leq(L+1/j){\left\|{g}\right\|}_{L^{p}((0,\infty),\theta d\theta)}.$
(40)

On the other hand, as $m$ is bounded and from Perseval’s formula, we have

${\left\|{F_{\lambda_{j}}}\right\|}_{L^{2}((0,\infty),\theta
d\theta)}\leq(L+1/j){\left\|{g}\right\|}_{L^{2}((0,\infty),\theta d\theta)}.$
(41)

From (40) and (41) there exists a subsequence of
$(\lambda_{j})_{j\in\mathbb{N}}$ denoted also $(\lambda_{j})_{j\in\mathbb{N}}$
such that the sequence $\\{F_{\lambda_{j}}\\}$ converge weakly to a function
$F$ in $L^{p}\cap L^{2}((0,\infty),\theta d\theta)$ and satisfying the
following inequality

${\left\|{F}\right\|}_{L^{p}((0,\infty),\theta d\theta)}\leq
L{\left\|{g}\right\|}_{L^{p}((0,\infty),\theta d\theta)}.$ (42)

Our purpose now is to show that
$F=\mathcal{H}_{0}\left(m(.)\mathcal{H}_{0}(g)\right)$ almost everywhere on
$(0,\infty).$
Let $N\geq 1$ and $\theta\in(0,\infty)$

$\displaystyle F_{\lambda}(\theta)$ $\displaystyle=$
$\displaystyle\chi_{(0,\lambda)}(\theta)\sum_{n=0}^{\infty}m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\right)a_{n}(G_{\lambda})\varphi_{n}\left(\cos(\frac{\theta}{\lambda})\right)$
$\displaystyle=$
$\displaystyle\chi_{(0,\lambda)}(\theta)\left[\sum_{n=0}^{N[\lambda]}+\sum_{n=N[\lambda]+1}^{\infty}\right]m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\right)a_{n}(G_{\lambda})\varphi_{n}\left(\cos(\frac{\theta}{\lambda})\right)$
$\displaystyle=$ $\displaystyle
F^{N}_{\lambda}(\theta)+K^{N}_{\lambda}(\theta).$

Using (9), the function
$F(\theta)=\varphi^{\alpha}_{n,c}\left(\cos(\theta)\right)$ satisfies the
following differential equation

$\displaystyle\mathcal{L}(F)(\theta)$ $\displaystyle=$
$\displaystyle-F^{\prime\prime}(\theta)-\frac{\cos(\theta)}{\sin(\theta)}F^{\prime}(\theta)+\left(c^{2}\cos^{2}(\theta)-\frac{1/4-\alpha^{2}}{\cos^{2}(\theta)}\right)F(\theta)$
$\displaystyle=$ $\displaystyle\chi_{n,\alpha}(c)F(\theta).$

Using the symmetry of $\mathcal{L}$ on $C^{\infty}_{c}(0,\infty)$, we obtain

$\displaystyle a_{n}(G_{\lambda})$ $\displaystyle=$
$\displaystyle{\left\langle{G_{\lambda},\varphi^{\alpha}_{n,c}}\right\rangle}_{L^{2}(0,1)}$
$\displaystyle=$
$\displaystyle\frac{1}{\chi_{n,\alpha}(c)}\int_{0}^{\frac{\pi}{2}}g_{\lambda}(\theta)\chi_{n,\alpha}(c)\varphi^{\alpha}_{n,c}\left(\cos(\theta)\right)\sin(\theta)d\theta$
$\displaystyle=$
$\displaystyle\frac{1}{\chi_{n,\alpha}(c)}\int_{0}^{\frac{\pi}{2}}g_{\lambda}(\theta)\mathcal{L}(F)(\theta)\sin(\theta)d\theta$
$\displaystyle=$
$\displaystyle\frac{\lambda^{2}}{\chi_{n,\alpha}(c)}\int_{0}^{\frac{\pi}{2}}\frac{1}{\lambda^{2}}\mathcal{L}\left(g_{\lambda}\right)(\theta)F(\theta)\sin(\theta)d\theta=\frac{\lambda^{2}}{\chi_{n,\alpha}(c)}a_{n}\left(\frac{1}{\lambda^{2}}\mathcal{L}\left(g_{\lambda}\right)\right).$

Using the previous equality, Perseval’s formula and the fact that $m$ is
bounded with the well known inequality
$\frac{2}{\pi}\theta\leq\sin(\theta)\leq\theta$, for
$0\leq\theta\leq\frac{\pi}{2}$ and (10), we obtain

$\displaystyle{\left\|{K_{\lambda}^{N}}\right\|}_{L^{2}((0,\infty),\theta
d\theta)}$ $\displaystyle=$
$\displaystyle\left[\int_{0}^{\infty}\chi_{(0,\lambda\frac{\pi}{2})}(\theta)\left|\sum_{n=N[\lambda]+1}^{\infty}m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\right)a_{n}(G_{\lambda})\varphi_{n}(\cos(\frac{\theta}{\lambda}))\right|^{2}\theta
d\theta\right]^{1/2}$ $\displaystyle=$
$\displaystyle\left[\int_{0}^{\lambda\frac{\pi}{2}}\left|\sum_{n=N[\lambda]+1}^{\infty}m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\right)a_{n}(G_{\lambda})\varphi_{n}(\cos\left(\frac{\theta}{\lambda})\right)\right|^{2}\theta
d\theta\right]^{1/2}$ $\displaystyle\leq$
$\displaystyle\sqrt{\frac{\pi}{2}}\left[\lambda\int_{0}^{\lambda\frac{\pi}{2}}\left|\sum_{n=N[\lambda]+1}^{\infty}m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\right)a_{n}(G_{\lambda})\varphi_{n}\left(\cos(\frac{\theta}{\lambda})\right)\right|^{2}\sin\left(\frac{\theta}{\lambda}\right)d\theta\right]^{1/2}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{\pi}{2}}\left[\lambda^{2}\sum_{n=N[\lambda]+1}^{\infty}m^{2}\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\right)a^{2}_{n}(G_{\lambda})\right]^{1/2}$
$\displaystyle\leq$ $\displaystyle
C\,\sqrt{\frac{\pi}{2}}\left[\frac{\lambda^{2}}{N^{4}}\sum_{n=N[\lambda]+1}^{\infty}a^{2}_{n}\left(\frac{1}{\lambda^{2}}\mathcal{L}(g_{\lambda})\right)\right]^{1/2}$
$\displaystyle\leq$
$\displaystyle\frac{C}{N^{2}}\,\sqrt{\frac{\pi}{2}}\left[{\left\|{g^{\prime\prime}+\frac{g^{\prime}}{\theta}}\right\|}_{L^{2}\left((0,\infty),\theta
d\theta\right)}+C{\left\|{g}\right\|}_{L^{2}\left((0,\infty),\theta
d\theta\right)}\right]$

Then we obtain ${\left\|{K^{N}_{\lambda}}\right\|}_{L^{2}((0,\infty),\theta
d\theta)}=O(\frac{1}{N^{2}})$ uniformly in $\lambda.$
Thus by the diagonal argument there exists a subsequence of
$\\{\lambda_{j}\\}$ noted again $\\{\lambda_{j}\\}$ such that for every $N\geq
1$, $\\{K^{N}_{\lambda_{j}}\\}_{j\in\mathbb{N}}$ converge weakly to a function
$K^{N}$ in $L^{2}((0,\infty),\theta d\theta)$ satisfy
${\left\|{K^{N}}\right\|}_{L^{2}((0,\infty),\theta
d\theta)}=O(\frac{1}{N^{2}})$, one conclude that there exists a sequence
$\\{N_{k}\\}$ such that $\\{K^{N_{k}}\\}_{k\in\mathbb{N}}$ converge to zero
almost everywhere on $(0,\infty)$. Let $F^{N_{k}}=F-K^{N_{k}}$, clearly we
have $\\{F_{{\lambda_{j}}}^{N_{k}}\\}_{j\in\mathbb{N}}$ converge weakly to
$F^{N_{k}}$ in $L^{2}(0,\infty)$ for every $k\in\mathbb{N}$. Moreover,
$\\{F^{N_{k}}\\}$ converge to $F$ almost everywhere on $(0,\infty).$
We prove now the following equality

$\lim_{j\to\infty}F_{{\lambda_{j}}}^{N_{k}}(x)=\int_{0}^{N_{k}}m(y)J_{0}(xy)\mathcal{H}_{0}(g)(y)ydy$
(43)

for every $x\in(0,\infty)$, the weak convergence of
$\\{F_{{\lambda_{j}}}^{N_{k}}\\}_{j\in\mathbb{N}}$ to $F^{N_{k}}$, in
particular,
${\left\langle{F_{{\lambda_{j}}}^{N_{k}},\chi_{(r,s)}}\right\rangle}$ converge
to ${\left\langle{F^{N_{k}},\chi_{(r,s)}}\right\rangle}$ for every
$0<r<s<\infty$ and by using the Lebesgue dominated convergence theorem which
give as ${\left\langle{F_{{\lambda_{j}}}^{N_{k}},\chi_{(r,s)}}\right\rangle}$
converge to
${\left\langle{\mathcal{H}_{\alpha}\left(\chi_{(0,N_{k}\pi)}m(.)\mathcal{H}_{\alpha}(g)\right),\chi_{(r,s)}}\right\rangle}$,
one conclude that
$F^{N_{k}}=\mathcal{H}_{\alpha}\left(\chi_{(0,N_{k}\pi)}m(.)\mathcal{H}_{\alpha}(g)\right)$
almost everywhere on $(0,\infty).$ Finally, as $k\to\infty$, we get our
purpose.
For the proof of (43), we need the uniform approximation of the family of
CPSWFs on $(0,1)$ which is given by the following estimates

$\varphi_{n,c}^{\alpha}(\cos(\frac{\theta}{\lambda}))=(-1)^{n}B_{n}\left(\cos(\frac{\theta}{\lambda})\right)^{\alpha+1/2}P_{n}^{(0,\alpha)}\left(\cos(\frac{2\theta}{\lambda})\right)+\gamma_{n}^{\alpha+1/2}O(\frac{c^{2}}{n})$
(44)

for every $\theta\in\mathbb{(}\lambda t_{n},\lambda\frac{\pi}{2})$, where
$t_{n}=\arccos(\gamma_{n})$ and
$\gamma_{n}\sim\frac{\sqrt{\alpha^{2}-1/4}}{\chi_{n,\alpha}^{1/2}(c)}$.

$\varphi_{n,c}^{\alpha}(\cos(\frac{\theta}{\lambda}))=A_{n}\,\chi^{1/4}_{n,\alpha}(c)\frac{\sqrt{S(\cos(\frac{\theta}{\lambda}))}J_{0}\left(\chi^{1/2}_{n,\alpha}(c)S(\cos(\frac{\theta}{\lambda}))\right)}{(\sin(\frac{\theta}{\lambda}))^{\frac{1}{2}}r_{n}\left(\cos(\frac{\theta}{\lambda})\right)^{1/4}}+R_{n}(\cos(\frac{\theta}{\lambda}))$
(45)

for every $\theta\in\mathbb{(}0,\lambda t_{n})$, where $A_{n}\sim 1,$
$r_{n}(t)=1-qt^{2}+\frac{1/4-\alpha^{2}}{\chi_{n,\alpha}^{1/2}(c)t^{2}}$ and
$\displaystyle\sup_{\theta\in\mathbb{(}0,t_{n})}\left|R_{n}(\cos(\theta))\right|\leq\frac{C}{\chi^{1/2}_{n,\alpha}(c)}$,
for more details see [17].
By a straightforward computation, we have

$\frac{\sqrt{S(\cos(\frac{\theta}{\lambda}))}}{(\sin(\frac{\theta}{\lambda}))^{1/2}r_{n}\left(\cos(\frac{\theta}{\lambda})\right)^{1/4}}=1-\beta(q)(1-\cos(\frac{\theta}{\lambda}))+o(1-\cos(\frac{\theta}{\lambda}))$

then, we can easily check that

$\varphi_{n,c}^{\alpha}(\cos(\frac{\theta}{\lambda}))=\chi_{n,\alpha}^{1/4}(c)J_{0}\left(\frac{\chi_{n,\alpha}^{1/2}(c)}{\lambda}\theta\right)+R_{n}(\cos(\frac{\theta}{\lambda})).$

Let $N>0$ and $\lambda>\max\\{\frac{2M}{\pi},N^{3}\\}$. By (45) we have, for
every $n\leq N[\lambda]$

$\displaystyle a_{n}(G_{\lambda})$ $\displaystyle=$
$\displaystyle{\left\langle{G_{\lambda},\varphi_{n,c}^{\alpha}}\right\rangle}_{L^{2}(0,1)}$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda}\int_{0}^{\lambda\frac{\pi}{2}}\left(g_{\lambda}\circ\arccos\right)(\cos(\frac{\theta}{\lambda}))\varphi_{n,c}^{\alpha}(\cos(\frac{\theta}{\lambda}))\sin(\frac{\theta}{\lambda})d\theta$
$\displaystyle=$
$\displaystyle\frac{\chi_{n,\alpha}^{1/4}(c)}{\lambda}\int_{0}^{\lambda\frac{\pi}{2}}J_{0}\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\theta\right)g(\theta)\sin(\frac{\theta}{\lambda})d\theta-\frac{\chi_{n,\alpha}^{1/4}(c)}{\lambda}\int_{\lambda
t_{n}}^{\lambda\frac{\pi}{2}}J_{0}\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\theta\right)g(\theta)\sin(\frac{\theta}{\lambda})d\theta$
$\displaystyle+$ $\displaystyle\frac{(-1)^{n}B_{n}}{\lambda}\int_{\lambda
t_{n}}^{\lambda\frac{\pi}{2}}g(\theta)P_{n}^{(0,\alpha)}\left(\cos(\frac{2\theta}{\lambda})\right)\left(\cos(\frac{\theta}{\lambda})\right)^{\alpha+1/2}\sin(\frac{\theta}{\lambda})d\theta+\frac{1}{n^{a}\chi^{1/4}_{n,\alpha}(c)}O(\frac{1}{\lambda^{b}})$
$\displaystyle=$
$\displaystyle\frac{\chi_{n,\alpha}^{1/4}(c)}{\lambda^{2}}\mathcal{H}_{0}(g)(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})+\frac{1}{n^{a}\chi^{1/4}_{n,\alpha}(c)}O(\frac{1}{\lambda^{b}}).$

where $a>1$ and $b>0.$ Indeed, using the fact that
$\sup_{x>0}|\sqrt{x}J_{\alpha}(x)|\leq C_{\alpha}$, see [23], and
$\lambda\sin(\frac{\theta}{\lambda})\leq\theta$ we have

$\displaystyle\left|\frac{\chi_{n,\alpha}^{1/4}(c)}{\lambda}\int_{\lambda
t_{n}}^{\lambda\frac{\pi}{2}}J_{0}\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\theta\right)g(\theta)\sin(\frac{\theta}{\lambda})d\theta\right|$
$\displaystyle\leq$ $\displaystyle\frac{1}{\lambda^{3/2}}\int_{\lambda
t_{n}}^{\lambda\frac{\pi}{2}}\left|\frac{\chi_{n,\alpha}^{1/4}(c)}{\lambda^{1/2}}J_{0}\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\theta\right)\right||g(\theta)|\theta
d\theta$ $\displaystyle\leq$
$\displaystyle\frac{1}{\lambda^{3/2}}\left[\int_{\lambda
t_{n}}^{\lambda\frac{\pi}{2}}\left|\frac{\chi_{n,\alpha}^{1/4}(c)\theta^{1/2}}{\lambda^{1/2}}J_{0}\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda}\theta\right)\right|^{2}\frac{d\theta}{\theta}\right]^{1/2}{\left\|{\theta
g}\right\|}_{L^{2}(0,\infty)}$ $\displaystyle\leq$
$\displaystyle\frac{C_{0}}{\lambda^{3/2}}(\ln(\frac{\pi}{2t_{n}}))^{1/2}{\left\|{\theta
g}\right\|}_{L^{2}(0,\infty)}$ $\displaystyle\leq$
$\displaystyle\frac{C_{0}}{\lambda^{3/2}\chi_{n,\alpha}^{1/4}(c)}{\left\|{\theta
g}\right\|}_{L^{2}(0,\infty)}.$

Moreover, using the fact that
$\left|P_{n}^{(0,\alpha)}(\cos(\frac{2\theta}{\lambda}))\right|\leq
P_{n}^{(0,\alpha)}(1)=O(n^{\alpha})$ and the deacreasing cosinus function with
$|B_{n}|=O(n^{1/2})$, we obtain

$\left|\displaystyle\frac{(-1)^{n}B_{n}}{\lambda}\int_{\lambda
t_{n}}^{\lambda\frac{\pi}{2}}g(\theta)P_{n}^{(0,\alpha)}\left(\cos(\frac{2\theta}{\lambda})\right)\left(\cos(\frac{\theta}{\lambda})\right)^{\alpha+1/2}\sin(\frac{\theta}{\lambda})d\theta\right|$

$\displaystyle\leq$ $\displaystyle\frac{|B_{n}|}{\lambda^{2}}\int_{\lambda
t_{n}}^{\lambda\frac{\pi}{2}}\left|g(\theta)\right|\left|P_{n}^{(0,\alpha)}\left(\cos(\frac{2\theta}{\lambda})\right)\right|\left(\cos(\frac{\theta}{\lambda})\right)^{\alpha+1/2}\theta
d\theta$ $\displaystyle\leq$
$\displaystyle\frac{C}{\lambda^{2}\chi^{1/4}_{n,\alpha}(c)}{\left\|{\theta^{3/2}g}\right\|}_{L^{2}(0,\infty)}$

Finally, there exist a constant $a>1$ and $b>0$ such that

$a_{n}(G_{\lambda})=\frac{\chi_{n,\alpha}^{1/4}(c)}{\lambda^{2}}\mathcal{H}_{0}(g)(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda})+\frac{1}{n^{a}\chi^{1/4}_{n,\alpha}(c)}O(\frac{1}{\lambda^{b}}).$
(46)

Hence, we obtain
$F_{{\lambda_{j}}}^{N_{k}}(\theta)=\chi_{(0,\frac{\pi}{2}\lambda_{j})}(\theta)\displaystyle\sum_{n=0}^{N_{k}[\lambda_{j}]}m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda_{j}}\right)a_{n}(G_{\lambda_{j}})\varphi^{\alpha}_{n,c}(\cos(\frac{\theta}{\lambda_{j}}))$

$\displaystyle=$
$\displaystyle\sum_{n=0}^{N_{k}[\lambda_{j}]}m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda_{j}}\right)a_{n}(G_{\lambda_{j}})\left(\chi_{(0,\lambda_{j}t_{n})}(\theta)\varphi_{n}(\cos(\frac{\theta}{\lambda_{j}}))+\chi_{(\lambda_{j}t_{n},\frac{\pi}{2}\lambda_{j})}(\theta)\varphi_{n}(\cos(\frac{\theta}{\lambda_{j}}))\right)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{N_{k}[\lambda_{j}]}m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda_{j}}\right)\left(\frac{\chi_{n,\alpha}^{1/4}(c)}{\lambda^{2}_{j}}\mathcal{H}_{0}(g)(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda_{j}})+\frac{1}{n^{a}\chi^{1/4}_{n,\alpha}(c)}O(\frac{1}{\lambda_{j}^{b}})\right)\chi_{(0,\lambda_{j}\frac{\pi}{2})}(\theta)\chi_{n,\alpha}^{1/4}(c)J_{0}\left(\frac{\chi_{n,\alpha}^{1/2}(c)}{\lambda}\theta\right)$
$\displaystyle+$
$\displaystyle\sum_{n=0}^{N_{k}[\lambda_{j}]}m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda_{j}}\right)a_{n}(G_{\lambda_{j}})\chi_{(\lambda_{j}t_{n},\frac{\pi}{2}\lambda_{j})}(\theta)\left((-1)^{n}B_{n}\left(\cos(\frac{\theta}{\lambda})\right)^{\alpha+1/2}P_{n}^{(0,\alpha)}\left(\cos(\frac{2\theta}{\lambda})\right)-\chi_{n,\alpha}^{1/4}(c)J_{0}\left(\frac{\chi_{n,\alpha}^{1/2}(c)}{\lambda}\theta\right)\right)$
$\displaystyle+$ $\displaystyle O(\frac{1}{\lambda^{\varepsilon}_{j}})$
$\displaystyle=$
$\displaystyle\chi_{(0,\frac{\pi}{2}\lambda_{j})}(\theta)\frac{1}{\lambda_{j}}\sum_{n=0}^{N_{k}[\lambda_{j}]}m\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda_{j}}\right)\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda_{j}}J_{0}\left(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda_{j}}\theta\right)\mathcal{H}_{0}(g)(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda_{j}})+O(\frac{1}{\lambda^{\varepsilon}_{j}}).$

where $\varepsilon>0.$ Indeed, from [29], we have

$\displaystyle(-1)^{n}B_{n}\left(\cos(\frac{\theta}{\lambda})\right)^{\alpha+1/2}P_{n}^{(0,\alpha)}\left(\cos(\frac{2\theta}{\lambda})\right)$
$\displaystyle=$
$\displaystyle(2n+\alpha+1)^{1/2}\left(\cos(\frac{\theta}{\lambda})\right)^{1/2}\left(\frac{\theta/\lambda}{\sin(\theta/\lambda)}\right)^{1/2}J_{0}\left(2(n+\frac{\alpha+1}{2})\frac{\theta}{\lambda}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{\lambda^{1/2}}O(\frac{(2\theta)^{1/2}}{n})$
$\displaystyle=$
$\displaystyle(2n+\alpha+1)^{1/2}J_{0}\left((2n+\alpha+1)\frac{\theta}{\lambda}\right)+O(\frac{1}{n}),$

and by using (10), one gets $\chi_{n,\alpha}(c)\sim(2n+\alpha+1)^{2}$, and
concludes that

$(-1)^{n}B_{n}\left(\cos(\frac{\theta}{\lambda})\right)^{\alpha+1/2}P_{n}^{(0,\alpha)}\left(\cos(\frac{2\theta}{\lambda})\right)-\chi_{n,\alpha}^{1/4}(c)J_{0}\left(\frac{\chi_{n,\alpha}^{1/2}(c)}{\lambda}\theta\right)=O(\frac{1}{n}).$

Further, from [5], we have
${\left\|{\varphi^{\alpha}_{n,c}}\right\|}_{L^{\infty}(0,1)}=O(\chi^{1/2}_{n,\alpha}(c))$,
then we obtain
$a_{n}(G_{\lambda_{j}})=O(\frac{\chi^{1/2}_{n,\alpha}(c)}{\lambda^{2}}).$
Finally, as $j\to\infty$, we get
$F^{N_{k}}=\mathcal{H}_{0}\left(\chi_{(0,N_{k})}m(.)\mathcal{H}_{0}(g)\right).$
∎

## References

  * [1] W.O. Amrein, A. M. Hinz and D. B. Pearson, Sturm-Liouville Theory: Past and Present, Birkhäuser, Basel-Boston-Berlin, (2005).
  * [2] A. I. Aptekarev, V. S. Buyarov & I. S. Degeza, Asymptotic behavior of the $L^{p}$-norms and the entropy for general orthogonal polynomials. Russian Acad. Sci. Sb. Math. 82 (1995), 373–395.
  * [3] J.J. Betankor, K.Stempak , Relating multipliers and transplantation for Fourier Bessel expansions and Hankel transform. Tohoku Math. J. 53 (2001), 109–129.
  * [4] A.Bonami and A. Karoui , Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions. Constr.Approx. 43 (2016), 15–45.
  * [5] M. Boulsane and A. Karoui, The Finite Hankel Transform Operator: Some Explicit and Local Estimates of the Eigenfunctions and Eigenvalues Decay Rates, J. Four. Anal. Appl, 24 , (2018), 1554-1578.
  * [6] M. Boulsane, P. Jaming and A. Souabni, Mean convergence of prolate spheroidal series and their extensions, J. Functional Analysis, 277, (2019)
  * [7] S. Bochner , Summation of multiple Fourier series by spherical means. Trans. Amer. Math. Soc 40 (1936), 175–207.
  * [8] V. Casarino and M. M. Peloso, $L^{p}$-summability of Riesz means for the sublaplacian on complex spheres, J. London Math. Soc. (2) 83 (2011) 137–152.
  * [9] P.Chen, X. T. Duong, D. He, S. Lee and L. Yan, Almost everywhere convergence of Bochner-Riesz means for the Hermite operator. Arxiv:2006.05689v3.
  * [10] Ó. Ciaurri and L. Roncal, The Bochner–Riesz means for Fourier–Bessel expansions Journal of Functional Analysis 228 (2005), 83–113.
  * [11] Ó.Ciaurri and L.Roncal, The Bochner–Riesz means for Fourier–Bessel expansions: Norm inequalities for the maximal operator and almost everywhere convergence. Journal of Approximation Theory 167 (2013), 121–146.
  * [12] S.Ciaurri and J. L. Varona, An Uniform Boundedness for Bochner-Riesz Operators Related to the Hankel Transform. J of lnequal and AppL, 2002, 7(6), 759–777.
  * [13] L. Colzani and G. Travaclini, Estimates for Riesz Kernels of Eigenfunction Expansions of Elliptic Differential Operators on Compact Manifolds.Journal of Functional Analysis 96,( 1991) l-30
  * [14] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, (1989).
  * [15] J. Horvath, L’oeuvre mathématique de Marcel Riesz I. Cahiers du séminaire d’histoire des mathématiques, tome 3 (1982), 83-121.
  * [16] S. Igari, On The Multipliers of Hankel Transform. Tohoku Math. Journal. 24(1972), 201-206.
  * [17] A. Karoui and I. Mehrzi, Asymptotic behaviors and numerical computations of the eigenfunctions and eigenvalues associated with the classical and circular prolate spheroidal wave functions. Journal of Applied Mathematics and Computation 218 (2012), 10871–10888.
  * [18] A. Karoui and A. Souabni, Generalized Prolate Spheroidal Wave Functions: Spectral Analysis and Approximation of Almost Band-limited Functions. J. Four. Anal. Appl. 22 (2016), 383–412.
  * [19] A. Karoui and A. Souabni, Weighted finite fourier transform operator:Uniform approximations of the eigenfunctions, eigenvalues decay and behaviour. J.Sci.Comp. 71 (2) (2017), 547–570.
  * [20] G. Mauceri , Riesz means for the eigenfunction expansions for a class of hypo-elliptic differential operators. Ann. Inst. Fourier 31 (1981), 115–140.
  * [21] C. Meaney, Divergent Cesaro and Riesz means of Jacobi and Laguerre expansions.Proceedings of the American Mathematical Society. 131 (2003), 3123–3128.
  * [22] D. Müller, On Riesz means of eigenfunction expansions for the Kohn Laplacian. J. Reine Angew. Math 401 (1989), 113–121.
  * [23] A. YA. Olenko. Upper bound on $\sqrt{x}J_{\mu}(x)$ and its applications. Integral Transforms and Special Functions. 17 (2006), 455–467.
  * [24] M. Riesz, Sur les fonctions conjugées. Math. Z. 27 (1927), 218–244.
  * [25] D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty I, Bell System Tech. J. 40 (1961), 43–64.
  * [26] D.Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty–IV: Extensions to many dimensions; generalized prolate spheroidal functions, Bell System Tech. J. 43 (1964), 3009–3057.
  * [27] D. Slepian, Some comments on Fourier analysis, uncertainty and modeling. SIAM Rev. 25 (1983) 379-393.
  * [28] L. Song , J. Xiao and X. Yan, Preduals of quadratic campanato spaces associated to operators with heat kernel bounds. J. Potential Analysis 41,(2014) 849–867.
  * [29] G. Szegö, Orthogonal polynomials, Fourth edition, American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., 1975.
  * [30] L. L. Wang and J. Zhang, A new generalization of the PSWFs with applications to spectral approximations on quasi-uniform grids. Appl. Comput. Harmon. Anal. 29 (2010), 303–329.