1 Introduction

The prolate spheroidal wave functions (PSWFs for short), were first known as the bounded eigenfunctions of the following differential operator \(L_{c},\) see [1]

$$\begin{aligned} L_{c}\varphi (t)=(1-t^{2})\frac{d^{2}}{dt^{2}}\varphi (t)-2t\frac{d}{dt} \varphi (t)-c^{2}t^{2}\varphi (t), \end{aligned}$$
(1.1)

where \(c>0\) is a real number.

At the beginning of the sixties of the twentieth century, Slepian and his collaborators have remarked that the PSWFs can be defined as the solution of the energy concentration problem of a Fourier bandlimited function on the interval \((-1,1 )\), see [2]. That is, the Fourier bandlimited function that maximizes the ratio given in by

$$\begin{aligned} \left( \alpha (f)\right) ^2=\displaystyle {\frac{\int _{-1}^{1}|f(t)|^2dt}{\int _{-\infty }^{+\infty }|f(t)|^2dt}}, \end{aligned}$$
(1.2)

is the first PSWFs, \(\varphi _{0,c}\). The second PSWFs, \(\varphi _{1,c}\), is the Fourier bandlimited function of unit 2-norm, perpendicular to \(\varphi _{0,c}\) that maximizes \((\alpha (f))^2)\). The nth PSWFs, \(\varphi _{n-1,c}\) is the Fourier bandlimited function of unit 2-norm, perpendicular to \({{span}}\{\varphi _{0,c},\varphi _{1,c},\ldots ,\varphi _{n-2,c}\}\) that maximizes \((\alpha (f))^2\). For more details about these functions the reader is referred to the following papers, see [2,3,4].

The most important property of PSWFs that distinguishes them from other special functions is their double orthogonality over finite interval \((-1,1)\) and infinite interval \((-\infty ,\infty )\). More precisely, the set of PSWFs is an orthogonal basis of \(L^2([-1,1],dx)\) and also an orthonormal basis of \(FB_c,\) where

$$\begin{aligned} FB_c=\left\{ f\in L^2({\mathbb {R}}):\,\mathrm {Supp}({\widehat{f}})\subset [-c,c]\right\} . \end{aligned}$$

Here \({\widehat{f}}\) denotes the Fourier transform of f. Due to their excellent properties, the PSWFs have found many applications in many areas such as communications, see [5] and in signal processing, see for example [6,7,8] . This explains the great interest of many authors for the computation of PSWFs and their corresponding eigenvalues, see for example [1, 9,10,11,12].

The first extension of the PSWFs is due to Slepian, in 1964. In fact he has generalized PSWFs to higher dimension \(D\ge 2,\) see [13]. He has showed that, in the special case where \(D=2,\) the radial part of the bi-dimensional PSWFs, called: circular prolate spheroidal wave functions (CPSWFs), are the eigenfunctions of the finite Hankel transform, i.e.

$$\begin{aligned} \widetilde{{\mathcal {H}}}_\nu \psi _{n}(t)=\int _0^{1}\sqrt{cts} J_\nu (cts)\psi _{n}(s)ds=\mu _{n}\psi _{n}(t),\quad t\in (0,+\infty ). \end{aligned}$$
(1.3)

Here \(J_\nu \) is the Bessel’s function of the first kind and order \(\nu >-1.\)

As it was done for the PSWFs, we have showed in [14] that the CPSWFs can be defined as the maximum energy concentration of a Hankel bandlimited function on the interval (0, 1). That is the Hankel bandlimited function that maximizes the ratio \((\beta (f))^2\), given by

$$\begin{aligned} \left( \beta (f)\right) ^2=\displaystyle {\frac{\int _{0}^{1}|f(t)|^2dt}{\int _{0}^{+\infty }|f(t)|^2dt}}, \end{aligned}$$
(1.4)

is the first CPSWFs, \(\psi _{0,c}^\nu \). The second CPSWFs, \(\psi _{1,c}^\nu \), is the Hankel bandlimited function of unit 2-norm, perpendicular to \(\psi _{0,c}^\nu \) that maximizes \((\beta (f))^2)\). The nth CPSWFs, \(\psi _{n-1,c}^\nu \) is the Hankel bandlimited function of unit 2-norm, perpendicular to \({{{\text{ span }}}}\{\psi _{0,c}^\nu ,\psi _{1,c}^\nu ,\ldots ,\psi _{n-2,c}^\nu \}\) that maximizes \((\beta (f))^2\).

Recently, we proved that CPSWFs have similar properties with PSWFs, see [15]. CPSWFs are used in many areas such as optics, see for example [16], and astrophysics see for example [17]. Their computation is done by several methods, see for example [13, 15, 18].

In the literature there several extensions of the Slepian’s functions, see [19,20,21,22]. To the best of our knowledge, in the literature, except [23], no one has showed that generalized Slepian’s functions solve a similar concentration problem as in (1.2) and (1.4).

In this paper we show that the extended Slepian’s functions solve the energy concentration which was solved first by Zayed [23]. Note here that our proof is different than the Zayed’s proof in [23]. Furthermore we show that Slepian’s functions are very good tool to approximate K bandlimited and essentially time-limited functions, where K is a reproducing kernel.

The outline of the paper is as follows: Sect. 2 is devoted to recall some basic fact of the GPSWFs and their properties which will be used frequently later. In Sect. 3 we show that Slepian’s functions are solutions of the energy maximization problem of K bandlimited functions. In Sect. 4 we use Slepian’s functions to approximate the K bandlimited and essentially time limited functions.

2 Preliminaries

In this section, we recall the definition of the GPSWFs and some of their properties which will be used frequently later, see [21].

Firstly, let E denote an arbitrary set of \({\mathbb {R}}\) and \(\omega (\cdot )\) denote a weight function on E. We consider an \(\omega \)-measurable interval \(T=(a,b)\subset E\) and the weighted \(L^{2}_{T,\omega }\)-space of all measurable functions f satisfying the following condition

$$\begin{aligned} \int _{T}|f(t)|^{2}\omega (t)dt<\infty . \end{aligned}$$

It is well-known that the weighted \(L^{2}_{T,\omega }\)-space is an Hilbert space.

Let k(t, s) denote a fixed real-valued function continuous on \(T\times E,\) such that

$$\begin{aligned} k(., s) \in L^{2}_{T,\omega }\quad \text{ for } \text{ any } s\in E, \end{aligned}$$

and \(k(t, s)=k(s,t)\) for \((t,s)\in T\times T.\) Note here that the condition of the symmetry of k(t, s) on \( T\times T\) is forgotten in [21]

We assume also that \(\{k(t,s),\;\;s\in E\}\) is complete in \(L^{2}_{T,\omega }\) and there exists a sequence \((s_n)_{n\in {\mathbb {N}}}\) such that the family \(\{k(\cdot ,s_n)\}_{n\in {\mathbb {N}}}\) is an orthormal basis of \(L^{2}_{T,\omega }.\)

Let \({\mathcal {F}}(E)\) denotes the linear space of all complex-valued functions defined on E. Consider the linear mapping \({\mathcal {L}}: L^{2}_{T,\omega }\rightarrow {\mathcal {F}}(E)\) defined by

$$\begin{aligned} {\mathcal {L}}_{f}(s)=\langle f,k(\cdot , s)\rangle = \int _T f(t)k(t, s)\omega (t)dt,\quad \text{ for } \text{ all } f \in L^2_{T, \omega }. \end{aligned}$$
(2.1)

We introduce the reproducing kernel \(K(\cdot , \cdot )\) expressed via

$$\begin{aligned} K(t, s) = \int _T k(r, t)k(r, s)\omega (r)dr \end{aligned}$$
(2.2)

and we suppose, in the sequel, that the kernel k is analytic. The set of all functions \({\mathcal {H}}_K=\{f\in L^{2}_{E,\omega };\ f(s) = \int _T g(t)k(t,s)\omega (t)dt;\;\;\;g\in L^{2}_{T,\omega }\}\) is a reproducing-kernel Hilbert space (RKHS) with the reproducing kernel \(K(\cdot , \cdot ).\)

Hereafter, all functions of the form (2.1) will be called K-bandlimited functions. We denote also by \({\mathcal {K}}\) the following operator

$$\begin{aligned} {\mathcal {K}}f( s) = \int _T K(s, t)f(t)\omega (t)dt. \end{aligned}$$
(2.3)

We assume also that the eigenvalues of the integral operator \({\mathcal {K}}={\mathcal {L}}\circ {\mathcal {L}}^*\) are simple. This is the case when \({\mathcal {L}}\) is one of the following integral operator: the finite Fourier transform [2], the finite Hankel transform [15], the finite weighted Fourier transform [24], the finite Dunkl transform [25], the Airy’s integral transform [19].

The Slepian’s functions, called sometimes generalized prolate spheroidal wave functions (GPSWFs), are the solutions of the Fredholm integral equation

$$\begin{aligned} \int _T \varphi _n(t)k(t, s)\omega (t)dt =\displaystyle {\gamma _n} \varphi _n(s),\quad s\in T, \end{aligned}$$
(2.4)

or equivalently

$$\begin{aligned} \int _T \varphi _n(t)K(t, s)\omega (t)dt =\displaystyle {\gamma _n^2} \varphi _n(s),\quad s\in T. \end{aligned}$$
(2.5)

In the sequel, we adopt the following normalization of the \(n{\mathrm{th}}\) GPSWFs

$$\begin{aligned} \Vert \varphi _n\Vert _{2,\omega ,T}^2=\int _T|\varphi _n(t)|^2\omega (t)dt=\displaystyle {\gamma _n^2}. \end{aligned}$$
(2.6)

In the following Theorem, borrowed from [21], we list some properties of the GPSWFs which will be used frequently in the sequel.

Theorem 2.1

Under the above notations and assumptions, we have

  1. (i)

    The n th GPSWFs \(\varphi _n\) can be naturally extended to functions \(\Phi _n\) defined on E such that \(\Phi _n\in {\mathcal {H}}_K\) and

    $$\begin{aligned} \int _T \Phi _m(t)\Phi _n(t)\omega (t)dt = \int _T \varphi _m(t)\varphi _n(t)\omega (t)dt =\gamma _n^2\delta _{m,n}; \end{aligned}$$
  2. (ii)

    The functions \(\Phi _n\) satisfy the following orthonormalization:

    $$\begin{aligned} \text{ For } \text{ all } \text{ integers } \text{ m } \text{ and } \text{ n, }\quad \int _E \Phi _m(t)\Phi _n(t)\omega (t)dt = \delta _{m,n}; \end{aligned}$$

In the sequel, we adopt the following definition: A function f is said to be time-limited to an interval \(T=(a,b)\) if \(f(x)=Df(x).\) Here \(Df(x)=f(x)\chi _{(a,b)}(x),\) where \(\chi _{T}(x)=\left\{ \begin{array}{cc} 1&{}x\in T\\ 0&{} x\not \in T. \end{array} \right. \)

3 Slepian’s functions as solution of energy concentration problem

The problem of energy concentration was solved first by Slepian, see [2], for Fourier bandlimited functions. A similar problem was solved in [14] for Hankel bandlimited functions. Recently, Zayed solved the problem of energy concentration in the reproducing kernel Hilbert space, see [23]. In this section, we solve the same problem for K bandlimited functions, differently. To do so, let f be a K bandlimited function and let \(\alpha (f)\) be the measure of its energy concentration on \(T=(a,b)\) defined by

$$\begin{aligned} \left( \alpha (f)\right) ^2=\displaystyle {\frac{\int _{T}|f(t)|^2\omega (t)dt}{\int _{E}|f(t)|^2\omega (t)dt}}. \end{aligned}$$
(3.1)

Remark 1

\(\diamond \) :

Note here that for any \(f(t)\in {\mathcal {D}}=\{f\in L^2_{E,\omega },\;\;f(x)=Df(x)\},\) we have \((\alpha (f))^2=1.\)

\(\diamond \) :

Since the kernel k is analytic, a nontrivial K-bandlimited signal cannot be time-limited. Hence, for any \(f(t) \in {\mathcal {H}}_K,\) we have \((\alpha (f))^2<1.\)

One can ask the following natural question: What is, if it exists, the maximum value of \(\alpha (f)^2,\) for \(f(t)\in {\mathcal {H}}_K\)? A second question can be asked, if the first one has a positive answer: This maximum value of \(\alpha (f)^2,\) is attained for each functions?

We remember firstly that the \(\displaystyle {\sup _{f\in {\mathcal {H}}_K}\alpha (f)^2}\) is the greatest eigenvalue of \({\mathcal {K}}.\) To do so, we suppose, without loss of generality, \(f\in {\mathcal {H}}_K\) is of unit 2-norm. Then we have,

$$\begin{aligned} \left( \alpha (f)\right) ^2= & {} \sup _{f\in {\mathcal {H}}_K}{\int _{a}^{b}|f(t)|^2\omega (t)dt} \\= & {} \sup _{g\in L^2_{T,\omega }}{\int _{a}^{b}\left( \int _{a}^{b}g(y)k(t,y)\omega (y)dy\right) \left( \int _{a}^{b}g(z)k(t,z)\omega (z)dz\right) \omega (t)dt}\\= & {} \sup _{g\in L^2_{T,\omega }}{\int _{a}^{b}g(y)\omega (y)dy\int _{a}^{b}g(z)\omega (z)dz\int _{a}^{b}k(t,y)k(t,z)\omega (t)dt} \\= & {} \sup _{g\in L^2_{T,\omega }}{\int _{a}^{b}g(y)\omega (y)dy\int _{a}^{b}K(y,z)g(z)\omega (z)dz} \\= & {} \sup _{g\in L^2_{T,\omega }}{\left<{\mathcal {K}}g,g\right>}. \end{aligned}$$

From classical results of functional analysis we have

$$\begin{aligned} \sup _{g\in L^2_{T,\omega }}{\left<{\mathcal {K}}g,g\right>}=\gamma _0^2 \end{aligned}$$

where \(\gamma _0^2\) is the greatest eigenvalue of \({\mathcal {K}}.\) Hence, \(\left( \alpha (f)\right) ^2=\gamma _0^2.\) Furthermore, from the normalization of the GPSWFs, one can easily check that \(\alpha (\pm \varphi _0)^2=\gamma _0^2.\) In the sequel, we show that the K-bandlimited functions that maximize \(\left( \alpha (f)\right) ^2\) are \(\pm \varphi _0.\) To do so, let consider \(f\in {\mathcal {H}}_K\) of unit 2-norm. Then, the expansion of f following Slepian’s functions is given by

$$\begin{aligned} f(t)=\displaystyle {\sum _{n=0}^{+\infty }a_n\varphi _{n}(t)}, \end{aligned}$$
(3.2)

where \(a_n=\frac{1}{\gamma _n^2}\int _{T}f(t)\varphi _{n}(t)\omega (t)dt.\)

Combining (2.6) together with (3.2) and the fact that \(f\in {\mathcal {H}}_K\) is of unit 2-norm, one obtains the following expression of the concentration energy measure

$$\begin{aligned} (\alpha (f))^2=\displaystyle {\sum _{n=0}^{+\infty }a_n^2\gamma _n^2}. \end{aligned}$$

Hence, the problem of energy concentration is equivalent to the following one: find a sequence \((a_n)_{n\in {\mathbb {N}}}\) satisfying the following conditions

(\(C_1\)):

\((\alpha (f))^2=\displaystyle {\sum _{n=0}^{+\infty }a_n^2\gamma _n^2}\) is maximal.

(\(C_2\)):

\(\displaystyle {\sum _{n=0}^{+\infty }a_n^2=1}\).

Note that the previous conditions together give

$$\begin{aligned} (\alpha (f))^2= & {} {{ \displaystyle {\sum _{n=0}^{+\infty }a_n^2\gamma _n^2}}}\nonumber \\= & {} {{a_0^2\gamma _0^2+ \displaystyle {\sum _{n=1}^{+\infty }a_n^2\gamma _n^2}}}\nonumber \\= & {} {{\left( \displaystyle {\sum _{n=0}^{+\infty }a_n^2}-\displaystyle {\sum _{n=1}^{+\infty }a_n^2}\right) \gamma _0^2+\displaystyle {\sum _{n=1}^{+\infty }a_n^2\gamma _n^2}}}\nonumber \\= & {} {{\left( 1-\displaystyle {\sum _{n=1}^{+\infty }a_n^2}\right) \gamma _0^2+\displaystyle {\sum _{n=1}^{+\infty }a_n^2\gamma _n^2}}}\nonumber \\= & {} \gamma _0^2+\displaystyle {\sum _{n=1}^{+\infty }a_n^2\left( \gamma _n^2-\gamma _0^2\right) }. \end{aligned}$$
(3.3)

Since the sequence \((\gamma _n^2)_{n\in {\mathbb {N}}}\) goes to 0 as n goes to \(\infty \), see [26], then the maximum value of \(\left( \alpha (f)\right) ^2\) is attained whenever for all \(n\in \mathbb {N^*},\) \(|a_n|= 0\) and \(|a_0|= 1.\) Hence, the functions \(f\in {\mathcal {H}}_K,\) that maximize \((\alpha (f))^2\) are \(\pm \varphi _0.\)

Moreover, the second Slepian’s function, \(\varphi _{1}\), is the K-bandlimited function of unit 2-norm, perpendicular to \(\varphi _{0}\) that maximizes \((\alpha (f))^2)\). The nth Slepian’s function, \(\varphi _{n}\) is the K-bandlimited function of unit 2-norm, perpendicular to \(span\{\varphi _{0},\varphi _{1},\ldots ,\varphi _{n-2}, {{\varphi _{n-1}}}\}\) that maximize \((\alpha (f))^2\).

4 Approximation of K-bandlimited and essentially time-limited functions by the use of Slepian’s functions

In this section we denote by \(E_{{\mathcal {H}}_K}(\epsilon _T)\) the following set of functions

$$\begin{aligned} E_{{\mathcal {H}}_K}(\epsilon _T)=\left\{ f\in {\mathcal {H}}_K,\;\;\Vert f\Vert _{2,{{\omega },E}}=1,\;\;\Vert f\chi _T\Vert _{2,{{\omega }},E}=1-\epsilon _T^2\right\} . \end{aligned}$$

Note here that in the special case where \(1-\epsilon _T^2<\gamma _0^2\) then their exists an infinite linearly independent functions that belongs to \(E_{{\mathcal {H}}_K}(\epsilon _T)\). In fact, for all \(k\in {\mathbb {N}}^*,\) the function \(f=a_N\varphi _{N}+a_{N+k}\varphi _{N+k}\) belongs to \(E_{{\mathcal {H}}_K}(\epsilon _T)\). Here \(a_N\) and \(a_{N+k}\) are solution of following system

$$\begin{aligned} \left\{ \begin{array}{lll} a_N^2+a_{N+k}^2=1\\ \gamma _N^2a_N^2+\gamma _{N+k}^2a_{N+k}^2=1-\epsilon _T^2. \end{array} \right. \end{aligned}$$

In this section, we assume that \(1-\epsilon _T^2<\gamma _0^2\) and we shall give the approximate dimension of \(E_{{\mathcal {H}}_K}(\epsilon _T)\). To do so, we need to introduce the following notations and definitions:

  1. (i)

    \(S_\psi ^N=\text{ Span }\{\psi _0, \psi _1,\ldots ,\psi _{N-1}\},\) where \(\psi _0\), \(\psi _1,\ldots ,\psi _{N-1}\) are N linearly independent functions.

  2. (ii)

    \(P_{S_\psi ^N}f\) is the orthogonal projection of f on \(S_\psi ^N.\)

  3. (iii)

    \(d(f,S_\psi ^N)=\Vert f-P_{S_\psi ^N}f\Vert \) is the distance between f and \(S_\psi ^N\).

Definition 4.1

We say that the approximate dimension of the set \(E_{{\mathcal {H}}_K}(\epsilon _T)\) is N, where \(N\in {\mathbb {N}}^{*}\), if there exists N linearly independent functions \(\psi _0\), \(\psi _1,\ldots ,\psi _{N-1}\) such that for all \(f\in E_{{\mathcal {H}}_K}(\epsilon _T),\) we have \(d(f,S_\psi ^N)\le \delta _N\), where \(\delta _N\) is small.

Definition 4.2

The deflection of \(E_{{\mathcal {H}}_{\mathcal {K}}}(\epsilon _T)\) from \(S_\psi ^N\subset L^2(E)\) is defined as

$$\begin{aligned} \delta _N(E_{{\mathcal {H}}_K}(\epsilon _T),S_\psi ^N):=\sup _{f\in E_{{\mathcal {H}}_K}(\epsilon _T)}d(f,S_\psi ^N). \end{aligned}$$
(4.1)

Remark 2

For any subspace \(S_\psi ^N\subset L^2(E)\), we have

$$\begin{aligned} \delta _N(E_{{\mathcal {H}}_{\mathcal {K}}}(\epsilon _T),S_\psi ^N)\le 1. \end{aligned}$$

The following lemma gives un upper bound of the deflection of \(E_{{\mathcal {H}}_{\mathcal {K}}}(\epsilon _T)\) from \(S_\varphi ^N=\text{ Span }\{\varphi _0, \varphi _1,\ldots ,\varphi _{N-1}\}.\) Here \(\varphi _k\) is the kth generalized Slepian’s function.

Lemma 4.3

The deflection of \(E_{{\mathcal {H}}_{\mathcal {K}}}(\epsilon _T)\) from \(S_{{{\varphi }}}^N\) satisfy the following inequality

$$\begin{aligned} (\delta _N(E_{{\mathcal {H}}_{\mathcal {K}}}(\epsilon _T),S_\varphi ^N))^2\le \frac{\epsilon _T^2}{1-\gamma _N^2}, \end{aligned}$$
(4.2)

for all \(\epsilon _T\) satisfying \(1-\epsilon _T^2< \gamma _0^2.\)

Proof

we shall prove firstly that the deflection of \(E_{{\mathcal {H}}_K}(\epsilon _T)\) from \(S_\varphi ^N\) is given by

$$\begin{aligned} (\delta _N(E_{{\mathcal {H}}_K}(\epsilon _T),S_\psi ^N))^2= \left\{ \begin{array}{lll} 1&{}\quad if&{}1-\epsilon _T^2\le \gamma _N^2\\ \displaystyle {\frac{\gamma _0^2-(1-\epsilon _T^2)}{\gamma _0^2-\gamma _N^2}}&{}\quad if&{}\gamma _N^2\le 1-\epsilon _T^2< \gamma _0^2. \end{array} \right. \end{aligned}$$
(4.3)

To prove (4.3), we consider the following two cases.

First case: Let \(\epsilon _T>0\) be a real number satisfying \(1-\epsilon _T^2<\gamma _N^2.\) Let \(a_N\) and \(a_{N+1}\) be a solution of following system

$$\begin{aligned} \left\{ \begin{array}{lll} a_N^2+a_{N+1}^2=1\\ \gamma _N^2a_N^2+\gamma _{N+1}^2a_{N+1}^2=1-\epsilon _T^2. \end{array} \right. \end{aligned}$$

Remark that in the special case where \(f=a_N\varphi _{N}+a_{N+1}\varphi _{N+1},\) we have \(f\in E_{{\mathcal {H}}_K}(\epsilon _T)\) and \(d(f,S_\varphi ^N)=1\). Consequently \(\delta _N(E_{{\mathcal {H}}_K}(\epsilon _T),S_\psi ^N)=1\).

Second case: Let suppose now that \( \gamma _N^2\le 1-\epsilon _T^2\le \gamma _0^2\). The Fourier expansion of \(f\in E_{{\mathcal {H}}_K}(\epsilon _T),\) with respect to the generalized Slepian’s functions is given by

$$\begin{aligned} f(t)=\displaystyle {\sum _{k\in \mathbf {N}}a_k\varphi _{k}(t)}. \end{aligned}$$
(4.4)

Here, \(\left( a_{k}\right) _{k\in {\mathbb {N}}}\) satisfy the following conditions

$$\begin{aligned} \displaystyle {\sum _{k\in \mathbf {N}}a_k^2}= & {} 1 \end{aligned}$$
(4.5)
$$\begin{aligned} \displaystyle {\sum _{k\in \mathbf {N}}a_k^2\gamma _k^2}= & {} 1-\epsilon _T^2. \end{aligned}$$
(4.6)

By multiplying both sides of (4.5) by \(\gamma _0^2\) and using (4.6), one gets

$$\begin{aligned} \gamma _0^2-(1-\epsilon _T^2)=\displaystyle {\sum _{k=0}^{N-1}(\gamma _0^2-\gamma _k^2)a_k^2}+\displaystyle {\sum _{k=N}^{+\infty }(\gamma _0^2-\gamma _k^2)a_k^2}. \end{aligned}$$
(4.7)

Note that from the Eq. (4.7) one obtains

$$\begin{aligned} \gamma _0^2-(1-\epsilon _T^2)\ge \displaystyle {\sum _{k=N}^{+\infty }(\gamma _0^2-\gamma _k^2)a_k^2}. \end{aligned}$$
(4.8)

Using the fact that the sequence \((\gamma _k^2)_k\) decays to zero, see [27], then (4.8) give us

$$\begin{aligned} \gamma _0^2-(1-\epsilon _T^2)\ge (\gamma _0^2-\gamma _N^2)\displaystyle {\sum _{k=N}^{+\infty }a_k^2}, \end{aligned}$$
(4.9)

or equivalently

$$\begin{aligned} \left(d(f,S_\varphi ^N)\right)^2\le \frac{\gamma _0^2-(1-\epsilon _T^2)}{\gamma _0^2-\gamma _N^2},\quad \forall f\in E_{{\mathcal {H}}_K}(\epsilon _T). \end{aligned}$$
(4.10)

To proceed further, we consider the function \(f=b_0\varphi _0+b_{N}\varphi _{N}\), where

$$\begin{aligned} \left\{ \begin{array}{lll} b_0^2+b_{N}^2=1\\ \gamma _0^2b_0^2+\gamma _{N}^2b_{N}^2=1-\epsilon _T^2. \end{array} \right. \end{aligned}$$

It is clear that the previous function f belongs to \(E_{{\mathcal {H}}_K}(\epsilon _T)\) and \(\left(d(f,S_\varphi ^N)\right)^2=b_N^2=\displaystyle {\frac{\gamma _0^2-(1-\epsilon _T^2)}{\gamma _0^2-\gamma _N^2}}\). Hence \(\delta _N(E_{{\mathcal {H}}_K}(\epsilon _T),S_\varphi ^N)=\displaystyle {\frac{\gamma _0^2-(1-\epsilon _T^2)}{\gamma _0^2-\gamma _N^2}}\). This achieve the first part of the proof. To complete the proof, we remark that, if \(\gamma _N^2<1-\epsilon _T^2<\gamma _0^2,\) then

$$\begin{aligned} \frac{\gamma _0^2-(1-\epsilon _T^2)}{\gamma _0^2-\gamma _N^2}<\frac{\epsilon _T^2}{1-\gamma _N^2}. \end{aligned}$$
(4.11)

In fact, if we note \(d=(\gamma _0^2-\gamma _N^2)(1-\gamma _N^2),\) then

$$\begin{aligned} \frac{\gamma _0^2-(1-\epsilon _T^2)}{\gamma _0^2-\gamma _N^2}-\frac{\epsilon _T^2}{1-\gamma _N^2}= & {} \frac{(\gamma _0^2-1+\epsilon _T^2)(1-\gamma _N^2)-(\gamma _0^2-\gamma _N^2)\epsilon _T^2}{d} \\= & {} \frac{\gamma _0^2-\gamma _0^2\gamma _N^2-1+\gamma _N^2+\epsilon _T^2-\epsilon _T^2\gamma _N^2-\epsilon _T^2\gamma _0^2+\epsilon _T^2\gamma _N^2}{d} \\= & {} \frac{\gamma _N^2(1-\gamma _0^2)-(1-\gamma _0^2)+\epsilon _T^2(1-\gamma _0^2)}{d} \\= & {} \frac{(1-\gamma _0^2)(\gamma _N^2+\epsilon _T^2-1)}{d} \end{aligned}$$

Since the sequence \((\gamma _n^2)\) goes to 0 when n goes to \(\infty \) and \(\gamma _0^2<1\) then \(d>0.\) Consequently if \(\gamma _N^2<1-\epsilon _T^2\) then \( \frac{\gamma _0^2-(1-\epsilon _T^2)}{\gamma _0^2-\gamma _N^2}-\frac{\epsilon _T^2}{1-\gamma _N^2}<0,\) which completes the proof of the equation (4.11) .

Moreover, if \(0<1-\epsilon _T^2\le \gamma _N^2,\) then

$$\begin{aligned} 1\le \frac{\epsilon _T^2}{1-\gamma _N^2}. \end{aligned}$$

Consequently, \(\delta _N(E_{{\mathcal {H}}_K}(\epsilon _T),S_\varphi ^N)\) satisfy the following inequality

$$\begin{aligned} (\delta _N(E_{{\mathcal {H}}_K}(\epsilon _T),S_\varphi ^N))^2\le \frac{\epsilon _T^2}{1-\gamma _N^2}, \end{aligned}$$
(4.12)

for all \(\epsilon _T\) satisfying \(1-\epsilon _T^2< \gamma _0^2\).