1 Introduction

Further Spectral Properties of the Weighted Finite Fourier Transform Operator and Related Applications.

NourElHouda Bourguiba and Ahmed Souabni 111 Corresponding author: Ahmed Souabni, email: souabniahmed@yahoo.fr
This work was supported in part by the Tunisian DGRST research grants UR 13ES47.

University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerte, Tunisia.

Abstract— In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWFs). These set of special functions have been introduced in  and  and they are defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator decay at a super-exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As a first application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a periodic Sobolev space. Our second application is related to a generalization in the context of the GPSWFs, of the Landau-Pollak approximate dimension of the space of band-limited and almost time-limited functions, given in the context of the classical PSWFs, the eigenfunctions of the finite Fourier transform operator. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

## 1 Introduction

We first recall that for , the classical prolate spheroidal wave functions (PSWFs) were first discovered and studied by D. Slepian and his co-authors, see [13, 14]. These PSWFs are defined as the solutions of the following energy maximization problem :

 Find g=argmaxf∈Bc∫1−1|f(t)|2dt∫R|f(t)|2dt

Where is the classical Paley-Winer space, defined by

 Bc={f∈L2(R),Support ˆf⊆[−c,c]}. (1)

Here, is the Fourier transform of defined by Moreover, it has been shown in , that the PSWFs are also the eigenfunctions of the integral operator defined on by

 Qcf(x)=∫1−1sin(c(x−y))π(x−y)f(y)dy

as well as the eigenfunctions of a commuting differential operator with and given by

 Lcf(x)=(1−x2)f′′(x)−2xf′(x)−c2x2f(x)

In this paper we are interested in a weighted family of PSWFs called the generalized prolate spheroidal wave functions (GPSWFs), recently given in  and . They are defined as the eigenfunctions of the Gegenbauer perturbed differential operator,

as well as the eigenfunctions of the following commuting weighted finite Fourier transform integral operator,

 F(α)cf(x)=∫1−1eicxyf(y)(1−y2)αdy,α>−1. (2)

The importance of PSWFs is due to their wide range of applications in different scientific research area such as signal processing, physics and applied mathematics. The question of quality of approximation by PSWFs has attracted some interests. In fact the authors in  have given an estimate of the decay of the PSWFs expansion of . Here is the Sobolev space over and of exponent . Later in  authors studied the convergence of the expansion of functions in a basis of PSWFs. We should mention that the problem of the best choice of the value of the bandwidth arises here. Numerical answers were given in . Recently in , the authors have given a precise answer to this question. Note that the convergence rate of the projection to f where or a periodic Sobolev space. This is done by using an estimate for the decay of the eigenvalues of the self-adjoint operator that is , where is the n-th eigenvalues of the integral operator (2). That’s why the first part of this work will be devoted to the study of the spectral decay of the eigenvalues of the integral operator .

On the other hand, in , the authors have shown that the PSWFs are well adapted for the approximation of functions that are band-limited and and also almost time-limited. That is for an this space is defined by

 Ec(ε)={f∈Bc; ∥f∥L2(R)=1,∥f∥2L2(I)≥1−ε}.

Here and is the Paley-Wiener space given by (1). In particular, the approximate dimension of the previous space of functions has been given in . Moreover, in , the author has extended this result in the context of the circular prolate spheroidal wave functions (CPSWFs). These laters are defined as the different eigenfunctions of the finite Hankel transform operator. In this paper we give the extension of the approximate dimension of the space of band-limited and almost time-limited function, in the context of the GPSWFs.

This work is organized as follows. In section 2, we give some mathematical preliminaries concerning the GPSWFs. Note that the extra weight function appearing in the weighted finite Fourier transform operator, generates new difficulties concerning the orthogonality of GPSWFs over Unlike the classical PSWFs, the GPSWFs are orthogonal over but their analytic extension to the real line are not orthogonal with respect to the Lebesgue measure. Hence, based on the properties of the GPSWFs, we give a procedure for recovering the orthogonality over of these laters. In section 3, we give some further estimates of the GPSWFs and some estimates of the eigenvalues of both integral and differential operators that define GPSWFs which will be useful later on in the study of the quality of approximations by GPSWFs. In section 4, we first extend to the case of the GPSWFs, the result given in , concerning the quality of approximation by the PSWFs of functions from the periodic Sobolev space Then, we show that the GPSWFs best approximate almost time-limited functions and belonging to the restricted Paley-Wiener space given by

 B(α)c={f∈L2(R),Support ˆf⊆[−c,c],ˆf∈L2((−c,c),ω−α(⋅c))}.

Finally, in section 5 we give some numerical examples that illustrate the different results of this work.

## 2 Mathematical Preliminaries

In this section, we give some mathematical preliminaries concerning some properties as well as the computation of the GPSWFs. These mathematical preliminaries are used to describe the different results of this work.

We first mention that some content of this paragraph has been borrowed from . Since, the GPSWFs are defined as the eigenfunctions of the weighted finite Fourier transform operator defined by

 F(α)cf(x)=∫1−1eicxyf(y)ωα(y)dy,ωα(y)=(1−y2)α,α>0, (3)

then, they are also the eigenfunctions of the operator defined on by

 Q(α)cg(x)=∫1−1c2πKα(c(x−y))g(y)ωα(y)dy,Kα(x)=√π2α+1/2Γ(α+1)Jα+1/2(x)xα+1/2. (4)

Here is the Bessel function of the first kind and order Moreover, the eigenvalues and of and are related to each others by the identity Note that the previous two integral operators commute with the following Gegenbauer-type Sturm-Liouville operator defined by

 L(α)c(f)(x)=−1ωα(x)ddx[ωα(x)(1−x2)f′(x)]+c2x2f(x),ωα(x)=(1−x2)α.

Also, note that the th eigenvalue of satisfies the following classical inequalities,

 n(n+2α+1)≤χαn(c)≤n(n+2α+1)+c2,∀n≥0. (5)

For more details, see . We will denote by the set of the eigenfunctions of and They are called generalized prolate spheroidal wave functions (GPSWFs). It has been shown that is an orthogonal basis of We recall that the restricted Paley-Wiener space of weighted band-limited functions has been defined in  by

 B(α)c={f∈L2(R),Support ˆf⊆[−c,c],ˆf∈L2((−c,c),ω−α(⋅c))}. (6)

Here, is the weighted space with norm given by

 ∥f∥2L2((−c,c),ω−α(⋅c))=∫c−c|f(t)|2ω−α(tc)dt.

Note that when the restricted Paley-Wiener space is reduced to the usual space Also, it has been shown that

 B(α)c⊆B(α′)c,∀α≥α′≥0. (7)

As a consequence, it has been shown in  that the eigenvalues decay with respect to the parameter that is for and any

 0<λαn(c)≤λα′n(c)<1,∀α≥α′≥0. (8)

Hence, by using the precise behaviour as well as the sharp decay rate of the given in , one gets a first result concerning the decay rate of the for any Moreover, since in , the authors have shown that for any and any , there exists such that :

 λ0n(c)

then by combining (8) and (9), one gets

 0<λαn(c)≤λ0n(c)

Note that a second decay rate of the and valid for has been recently given in . More precisely, it has been shown in , that if and , then there exist and a constant such that

 λαn(c)≤Cαexp(−(2n+1)[log(4n+4α+2ec)+Cαc22n+1]),∀n≥Nα(c). (11)

Also, by using the fact that

 ψ(α)n,c(x)=1μαn∫1−1ψ(α)n,c(t)(1−t2)αeitcxdt=1cμαn∫Rψ(α)n,c(t/c)(1−(t/c)2)αeitxdt,

then, we have

 ˆψ(α)n,c(x)=√2π√cλnψ(α)n,c(xc)(1−(xc)2)α1[−c,c](x). (12)

Note that the GPSWFs are normalized by the following rule,

 ∥ψ(α)n,c∥2L2(I,ωα)=∫1−1(ψ(α)n,c(t))2ωα(t)dt=λ(α)n(c). (13)

The extra weight function generates new complications concerning the orthogonality of over . To solve this problem, we propose the following procedure. We define on the following inner product

 α=12π∫Rˆf(x)ˆg(x)ω−α(xc)dx (14)

By using (12) and (13), we have :

 <ψ(α)n,c,ψ(α)m,c>α=1cλαn(c)∫c−cψ(α)n,c(xc)ψ(α)m,c(xc)ωα(xc)dx=δnm (15)

Here is the Kronecker’s symbol. For more details, see . Now for , we have for some . By using the Fourier inversion formula and the fact that is an orthogonal basis of one gets

 f(x) = 12π∫c−ceixyˆf(y)dy=12π∫c−ceixyg(y)ωα(yc)dy (16) = c2π∫1−1eicxyg(cy)ωα(y)dy=c2π∫1−1eicxy(∞∑n=0αnψ(α)n,c(y))ωα(y)dy = c2π∞∑n=0αn∫1−1ψ(α)n,c(y)eicxyωα(y)dy=c2π∞∑n=0αnμαn(c)ψ(α)n,c(x)

where Consequently, the set is an orthogonal basis of and an orthonormal basis of when this later is equipped with the inner product

Also, in , the authors have proposed the following scheme for the computation of the GPSWFs. In fact, since then its series expansion with respect to the Jacobi polynomials basis is given by

 ψ(α)n,c(x)=∑k≥0βnk˜P(α,α)k(x),x∈[−1,1]. (17)

Here, denotes the normalized Jacobi polynomial of degree given by

 ˜P(α,α)k(x)=1√hkP(α,α)k(x),hk=22α+1Γ2(k+α+1)k!(2k+2α+1)Γ(k+2α+1). (18)

The expansion coefficients as well as the eigenvalues this system is given by

 √(k+1)(k+2)(k+2α+1)(k+2α+2)(2k+2α+3)√(2k+2α+5)(2k+2α+1)c2βnk+2+(k(k+2α+1)+c22k(k+2α+1)+2α−1(2k+2α+3)(2k+2α−1))βnk (19) +√k(k−1)(k+2α)(k+2α−1)(2k+2α−1)√(2k+2α+1)(2k+2α−3)c2βnk−2=χαn(c)βnk,k≥0.

## 3 Further Estimates of the GPSWFs and their associated eigenvalues.

In this section, we first give some new estimates of the eigenvalues of the eigenvalues associated with the GPSWFs. Then, we study a local estimate of these laters.

### 3.1 Estimates of the eigenvalues.

We first prove in a fairly simple way, the super-exponential decay rate of the eigenvalues of the weighted Fourier transform operator. We should the techniques used in this proof is inspired from the recent paper , where a similar result has been given in the special case .

###### Proposition 1.

For given real numbers , and for any integer we have

 |μαn(c)|≤kαcα+32log(2n−1ec)(ec2n−1)n+α+12,kα=(2e)3+α2π5/4(Γ(α+1))1/2. (20)

Proof: We first recall the Courant-Fischer-Weyl Min-Max variational principle concerning the eigenvalues of a compact self-adjoint operator on a Hilbert space with positive eigenvalues arranged in the decreasing order then we have

 λn=minf∈Snmaxf∈S⊥n,∥f∥G=1G,

where is a subspace of of dimension In our case, we have We consider the special case of

 Sn=Span{P(α,α)0,P(α,α)1,…,P(α,α)n−1}

and

 f=∑k≥nakP(α,α)k∈S⊥n,∥f∥L2(I,ωα)=∑k≥n|ak|2=1.

From [ page 456], we have

 ∥FαcP(α,α)k∥L2(I,ωα)=∥√π(2cx)α+1/2Γ(k+α+1)Γ(k+1)Jk+α+1/2(cx)∥L2(I,ωα). (21)

and

 ∥xk∥2L2(I,ωα)=β(k+1/2,α+1)≤√πeΓ(α+1)kα+1. (22)

By using the well-known bound of the Bessel function given by

 |Jα(x)|≤|x|α2αΓ(α+1)∀α>−1/2,∀x∈R, (23)

one gets,

 ∥Fαc˜P(α,α)k∥L2(I,ωα) ≤ √πΓ(k+α+1)Γ(k+1)Γ(k+α+3/2)(c2)k∥xk∥L2(I,ωα). (24) ≤ √πΓ(k+α+1)Γ(k+1)Γ(k+α+3/2)(c2)k(πe)1/4√Γ(α+1)kα+12 (25)

Next, we use the following useful inequalities for the Gamma function, see 

 √2e(x+1/2e)x+1/2≤Γ(x+1)≤√2π(x+1/2e)x+1/2,x>0. (26)

Then, we have

 ∥Fαc˜P(α,α)k∥L2(I,ωα) ≤ π5/4√Γ(α+1)√ce3/41kα+22(ce2k+1)k+1/2. (27)

Hence, for the previous and by using Hölder’s inequality, combined with the Minkowski’s inequality for an infinite sums, and taking into account that so that for one gets:

 |L2(I,ωα)|=|L2(I,ωα)| ≤ ∑k≥n|ak|2∥Fαc˜P(α,α)k∥2L2(I,ωα) (28) ≤ (π5/4√Γ(α+1)√ce3/4∑k≥n(ec2k+1)k+1/21kα+22)2 (29)

The decay of the sequence appearing in the previous sum, allows us to compare this later with its integral counterpart, that is

 ∑k≥n(ec2k+1)k+1/21kα+22≤∫∞n−1e−(x+1/2)log(2x+1ec)xα+22dx≤∫∞n−1e−(x+1/2)log(2n−1ec)(n−1)α+22dx. (30)

Hence, by using (29) and (30), one concludes that

 maxf∈S⊥n,∥f∥L2(I,ωα)=11/2L2(I,ωα) ≤ π5/4√Γ(α+1)√ce341log(2n−1ec)(ec2n−1)n−1/21(n−1)α+22 (31) ≤ 2α+22π5/4√Γ(α+1)(ec)α+321log(2n−1ec)(ec2n−1)n+α+12 (32)

To conclude the proof of the theorem, it suffices to use the Courant-Fischer-Weyl Min-Max variational principle.

###### Remark 1.

By the fact that and straightforward computations one gets for and for all :

 λαn(c)≤Kαcα+2log2(2n−1ec)(ec2n−1)2n+α+1withKα=π3/22(2/e)α+3Γ(α+1)
###### Remark 2.

The decay rate given by the last proposition improve the two results mentioned above (10) and (11). Indeed, we have a more precise result furthermore the proof given in  contains two serious drawbacks. It is only valid in the particular case and clearly more sophisticated than the previous one.

### 3.2 Local estimates of the GPSWFs

In this paragraph, we give a precise local estimate of the GPSWFs, which is valid for Then, this local estimate will be used to provide us with a new lower bound for the eigenvalues of the differential operator for For this purpose, we first recall that are the bounded solutions of the following ODE :

 ωα(x)L(α)cψ(x)+wα(x)χn,αψ(x)=(wα(x)ψ′(x)(1−x2))′+wα(x)(χαn−c2x2)ψ(x)=0,x∈[−1,1]. (33)

As it is done in , we consider the incomplete elliptic integral

 S(x)=∫1x√1−qt21−t2dt,q=c2χαn(c)<1. (34)

Then, we write into the form

 ψ(x)=ϕα(x)V(S(x)),ϕα(x)=(1−x2)(−1−2α)/4(1−qx2)−1/4. (35)

By combining (33), (35) and using straightforward computations, it can be easily checked that satisfies the following second order differential equation

 V′′(s)+(χn,α+θα(s))V(s)=0,s∈[0,S(0)] (36)

with

 θα(S(x))=(wα(x)(1−x2)ϕ′α(x))′1ϕα(x)wα(x)(1−qx2).

Define then we have It follows that can be written as

 θα(S(x))=116(1−qx2)[(Q′α(x)Qα(x))2(1−x2)−4ddx((1−x2)Q′α(x)Qα(x))−4(1−x2)Q′α(x)Qα(x)w′α(x)wα(x)]. (37)

Since then we have and Hence, we have

 θα(S(x)) = θ0(S(x))+−14(1−qx2)[(w′α(x)wα(x))2(1−x2)+2ddx[(1−x2)w′α(x)wα(x)]] (38) = θ0(S(x))+1(1−x2)(1−qx2)(−α2x2+α(1−x2)) = θ0(S(x))+α(1+α)(1−qx2)−α2(1−qx2)(1−x2).

The previous equality allows us to prove the following lemma.

###### Lemma 1.

For any and , we have is increasing on .

Proof: We use the notation and with Straightforward computations, we have

 (θα(S(x)))′ = (θ0(S(x)))′+2x(1−qx2)2(1−x2)2[qα(α+1)(1−x2)2−qα2(1−x2)−α2(1−qx2)] = 2xH(u)4u2(1−q+qu)4

where

 H(u) = G(u)+4(1−q+qu2)2[q(α2+α)u2−α2(1−q+2qu)] ≥ (1−q+qu2)2[G(u)+4[q(α2+α)u2−α2(1−q+2qu)]]

with given by [LABEL:e] then

 H(u) ≥ (1−q+qu2)2[(1−q)24(4−4q+u(17q−3))+4[−qα3α+1−(1−q)α2]] ≥ (1−q+qu2)2[(1−q)3−4α2(1−q1α+1)+(1−q)24(17q−3)u].

Then if and we have

 H(u) ≥ (1−q+qu2)2[(1−q)3−4α2(1−q1α+1)+(1−q)24(17q−3)] ≥ (1−q+qu2)2[(1−q)24(1+13q)−4α2]≥0

As a consequence of the previous lemma, one gets.

###### Lemma 2.

For any and for any integer with , we have

 supx∈[0,1]√(1−x2)(1−qx2)ωα(x)|ψ(α)n,c(x)|2≤|