Bochner-Riesz Means Convergence of Prolate Spheroidal Series and Their Extensions
aa r X i v : . [ m a t h . C A ] J a n Bochner-Riesz Means Convergence of Sturm-Liouville EigenfunctionsSeries and related applications
Mourad Boulsane a & Ahmed Souabni a 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 thespectrum of the Sturm-Liouville operator in L p ([ a, b ] , ω ) . Our purpose is to establish suitableconditions under which the Bochner-Riesz expansion of a function f ∈ L p ([ a, b ] , ω ) convergesto f in L p ([ a, b ] , ω ). Then we apply this result in the case of Jacobi polynomials and twogeneralisations of Slepian’s basis. Keywords:
Bochner-Riesz mean convergence, eigenfunctions and eigenvalues, prolate spheroidal wavefunctions.
The subject of the L p -Bochner-Riesz mean convergence of orthogonal series, has attracted specialattention since several decades ago. This kind of convergence is briefly described as follows. Let 1 ≤ p < ∞ , a, b ∈ R and { ϕ n } an orthonormal set of eigenfunctions of a positive self-adjoint differentialoperator L associated with eigenvalues χ n on a weighted Hilbert space L ( I, dω ) − space, where ω is a positive bounded weight function. We define the expansion coefficients of f ∈ L p ([ a, b ] , ω ) by a n ( f ) = R ba f ( x ) ϕ n ( x ) ω ( x ) dx. The orthonormal set { ϕ n } is said to have the Bochner-Riesz meanconvergence of order p over the Banach space L p ( I, dω ) if for all f ∈ L p ( I, dω ) , we havelim R →∞ Z ba | f ( x ) − Ψ δR f ( x ) | p ω ( x ) dx = 0 , (1)where Ψ δR f = ∞ X n =0 (cid:16) − χ n R (cid:17) δ + a n ( f ) ϕ n . To the best of our knowledge, M. Riesz was the first in the late 1911’s, to investigate this problemin some special cases. Our problem is a modification summability method of Riesz mean intro-duced by Salomon Bochner given by (1) . In [5], S.Bochner started by studying this problem forthe trigonometric exponential case in higher dimension. Furthermore, in [11], the authors haveproved a Bochner-Riesz mean convergence for the orthonormal eigenvectors system of a second or-der elliptic differential operator on a compact N-dimensional manifold M for 1 ≤ p ≤ N +1 N +3 and δ > N (cid:12)(cid:12)(cid:12) p − (cid:12)(cid:12)(cid:12) − . Mauceri and M¨ u ller studied also in [16] and [18] this problem in the frameworkof the Heisenberg group.This problem has been analysed for Fourier-Bessel expansions series in [8]and [9]. Moreover, in [6], authors also solved this question in the case of sublaplacien on the sphere S n − in the complex n-dimensional space C n ,where it has been shown that we have convergencefor δ > (2 n − (cid:12)(cid:12)(cid:12) − p (cid:12)(cid:12)(cid:12) . The weak type convergence is investigated in this problem. Indeed, we say Corresponding author: Mourad Boulsane, Email: [email protected] ϕ n of L p ( I, ω ) have a weakly Bochner-Riesz mean convergence if Ψ δR f converge to f almost everywhere for every f ∈ L p ( I, ω ). This problem is solved in some special casesof orthonormal systems like Jacobi and Laguerre polynomials in [17] and for the eigenfunctions ofthe Hermite operator in higher dimension in [7].In this work, we extend the L p -Bochner-Riesz means convergence to the Jacobi polynomials , the cir-cular and the generalized prolate spheroidal wave functions denoted by (CPSWFs) and (GPSWFs),respectively. The two last families are defined respectively as the eigenfunctions of the operators H αc f ( x ) = Z √ cxyJ α ( cxy ) f ( y ) dy, F ( α ) c f ( x ) = Z − e icxy f ( y )(1 − y ) α dy, where α > − / , c > α = 0 . Our first aim in this paper is to prove that we have L p -Bochner-Riesz mean convergence of anorthonormal basis of eigenfunctions ϕ n of a Sturm-Liouville operator L associated with eigenvalues χ n on the weighted-space L p ( I, ω ) whenever δ > γ ( p ′ ) ε where γ ( p ) satisfy k ϕ n k p ≤ Cn γ ( p ) . Oursecond aim is to apply the previous result to solve the problem of Bochner Riesz related to thespectrum of the Sturm-Liouville operator for the two previous generalizations of PSWFs and for theJacobi polynomials.This work is organised as follows. In section 2, we give some mathematical preliminaries onSturm-Liouville theory and some properties of the CPSWfs and GPSWFs. Note that these func-tions can be considered as generalizations of the spherical Bessel functions j ( α ) n and Gegenbauer’spolynomials e P ( α ) n , respectively. In section 3, we formalize some conditions on the orthonormal basisof L ( I, ω ) that will ensure the convergence of Ψ δR .f to f in L p as R → ∞ . Finally, in section 4, weapply this general result to find a sufficient condition for the L p - Bochner-Riesz means convergencein the case of weighted and circular prolate spheroidal wave functions and consequently for Jacobipolynomials as a particular case of weighted prolates. Also, we further give a necessary condition inthe case of Jacobi polynomials. In this paragraph, we give some mathematical preliminaries that will be frequently used in the proofsof the different results of this work.
The Sturm-Liouville differential operator is defined as follows, see for example [1], L y ( x ) = ddx [ p ( x ) y ′ ( x )] + q ( x ) y ( x ) , x ∈ I = ( a, b ) . (2)with r = p , q ∈ L ( I, R ) . The Sturm-Liouville eigenvalues problem is given by the following differ-ential equation : L .u ( x ) = − χω ( x ) u ( x ) , σ ∈ L ( I, R ) . (3)That is ddx h p ( x ) dudx i + q ( x ) u ( x ) + χω ( x ) u ( x ) = 0 , x ∈ I. (4)2ote that a Sturm-Liouville operator satisfies the following properties,1. u L v − v L u = h p ( uv ′ − vu ′ ) i ′ ( Lagrange’s identity )2. The eigenvalues of L are real and form an infinite countable set χ < χ < · · · < χ n < · · · with lim n → + ∞ χ n = + ∞ .
3. For each eigenvalue χ n there exists an eigenfunction φ n having n zeros on [ a, b ] .
4. Eigenfunctions corresponding to different eigenvalues are orthogonal with respect to the fol-lowing inner product h f, g i ω = Z ba f ( x ) g ( x ) ω ( x ) dx, f, g ∈ L ( I, ω ) . In the sequel, we assume that ω ( x ) ≥
0, for x ∈ ( a, b ) . We first recall that, for c >
0, the prolate spheroidal wave functions PSWFs denoted by ψ n,c havebeen introduced by D.Slepian as solutions of the following energy maximization problemFind f = arg max f ∈ B c R − | f ( t ) | dt R R | f ( t ) | dt , where B c is the classical Paley-Wiener space, defined by B c = n f ∈ L ( R ) , Support b f ⊆ [ − c, c ] o . (5)Here, b f is the Fourier transform of f ∈ L ( R ) . It has been shown that they are also eigenfunctionsof the integral operator with sinc kernel. A breakthrough in the theory of Slepian functions is dueto 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 functionswe refer reader to [20, 21, 22]. 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 the2d-Slepian as some authors called, introduced by D.Slepian[22] as solutions of the following problemFind f = arg max f ∈ HB αc R | f ( t ) | dt R ∞ | f ( t ) | dt , where HB αc is the Hankel Paley-Wiener space, defined by HB αc = (cid:8) f ∈ L ( R ) , Support H α f ⊆ [ − c, c ] (cid:9) . (6)Here the Hankel transform H α is defined for f ∈ L (0 , ∞ ) by H α f ( x ) = Z ∞ √ xyJ α ( xy ) f ( y ) dy. Here J α ( . ) is the Bessel function and α > − /
2. Like Fourier transform, H α can be extended into aunitary operator on L (0 , ∞ ). They are also the different band-limited eigenfunctions of the finiteHankel transform H αc defined on L (0 ,
1) with kernel H αc ( x, y ) = √ cxyJ α ( cxy ) where J α is theBessel function of the first type and order α > − , see for example [3]. That is H αc ( ϕ αn,c ) = µ n,α ( c ) ϕ αn,c . (7)3n his pioneer work [22], D. Slepian has shown that the compact integral operator H αc commuteswith the following Sturm-Liouville differential operator L αc defined on C ([0 , L αc ( φ ) = − ddx (cid:20) (1 − x ) ddx φ (cid:21) + c x − − α x φ. (8)Hence, ϕ αn,c is the n − th order bounded eigenfunction of the positive self-adjoint operator L αc associated with the eigenvalue χ n,α ( c ) , that is − ddx (cid:20) (1 − x ) ddx ϕ αn,c ( x ) (cid:21) + c x − − α x ϕ αn,c ( x ) = χ n,α ( c ) ϕ αn,c ( x ) , x ∈ [0 , . (9)The orthonormal family ϕ αn,c form an orthonormal basis of L (0 ,
1) and the associated eigenvaluesfamily χ n,α ( c ) satisfy the following inequality, see [22](2 n + α + 1 / n + α + 3 / ≤ χ n,α ( c ) ≤ (2 n + α + 1 / n + α + 3 /
2) + c (10)The second family we consider in this work is the weighted, (some times called generalized), prolatespheroidal wave functions introduced by Wang-Zhang [25] as solutions of a Sturm-Liouville problemor equivalently eigenfunctions of an integral operator. GPSWFs are also solutions of the followingproblem as given in [15] Find f = arg max f ∈ B αc k f k L ωα ( I ) k b f k L ( ω − α ( · c )) , where ω α ( x ) = (1 − x ) α and B ( α ) c is the restricted Paley-Winer space, defined by B ( α ) c = { f ∈ L ( R ) , Support b f ⊆ [ − c, c ] , b f ∈ L (cid:0) ( − c, c ) , ω − α ( · c ) (cid:1) } . More precisely, the GPSWFs are the eigenfunctions of the weighted finite Fourier transform operator F ( α ) c defined by F ( α ) c f ( x ) = Z − e icxy f ( y ) ω α ( y ) d y. (11)It is well known, (see [15, 25]) that they are also eigenfunctions of the compact and positive operator Q ( α ) c = c π F ( α ) ∗ c ◦ F ( α ) c which is defined on L ( I, ω α ) by Q ( α ) c g ( x ) = Z − c π K α ( c ( x − y )) g ( y ) ω α ( y )y. (12)Here, K α ( x ) = √ π α +1 / Γ( α + 1) J α +1 / ( x ) x α +1 / It has been shown in [15, 25] that the last two integral operators commute with the followingSturm-Liouville operator L ( α ) c defined on C [ − ,
1] by L ( α ) c ( f )( x ) = − ddx (cid:2) ω α ( x )(1 − x ) f ′ ( x ) (cid:3) + c x ω α ( x ) f ( x ) . (13)Also, note that the ( n + 1) − th eigenvalue χ αn ( c ) of L ( α ) c satisfies the following classical inequalities, n ( n + 2 α + 1) ≤ χ αn ( c ) ≤ n ( n + 2 α + 1) + c , ∀ n ≥ . (14)4 Main Result
Let (
I, ω ) be a measured space such that ω is a bounded weight function. We denote by p ′ = pp − the dual index of p .Throughout this section, L denotes a Sturm-Liouville operator and ϕ n (respectively λ n ) thesequence of the associated eigenfunctions (respectively eigenvalues). The Riesz means of index δ > L of a function f ∈ C ∞ ( I, R ) are defined asΨ δR f = ∞ X n =0 (cid:16) − λ n R (cid:17) δ + a n ( f ) ϕ n with a n ( f ) = Z I f ( y ) ϕ n ( y ) dµ ( y ) . (15)Ψ δR f can be also written asΨ δR .f ( x ) = Z I K δR ( x, y ) f ( y ) dµ ( y ) where K δR ( x, y ) = ∞ X n =0 (cid:16) − λ n R (cid:17) δ + ϕ n ( x ) ϕ n ( y )Our aim in this section is to define several conditions on ϕ n that will ensure the convergence ofΨ δR .f to f in the L p norm as R → ∞ . Assume that ϕ n satisfies the following conditions :( A ) For every 1 ≤ p ≤ ∞ , every n , ϕ n ∈ L p ( I, ω ). Further, we assume that there is a constant γ ( p ) ≥ k ϕ n k L p ( µ ) ≤ Cn γ ( p ) .( B ) The sequence ( λ n ) of the eigenvalues of the operator L satisfies the following properties1. X λ n ∈ ( m,M ) ≤ C ( M − m ) for all 0 ≤ m < M .2. There exists ε > λ n ≥ Cn ε First of all, we start by giving sense of Ψ δR .f for every f ∈ L p ( µ ). Indeed, (cid:13)(cid:13) K δR (cid:13)(cid:13) L p ( µ ) ⊗ L p ′ ( µ ) ≤ X λ n Theorem 1. With the above notation and under conditions ( A ) and ( B ) with δ > δ ( p ) = max { γ ( p ′ ) ε , } ,there exists a constant C > satisfying the following inequality (cid:13)(cid:13) Ψ δR (cid:13)(cid:13) ( L p ( I,w ) ,L p ( I,w )) ≤ C. (16)The following lemma will be used in the proof of the previous theorem. Lemma 1. Let ≤ p ≤ then for every f ∈ L p ( I, ω ) , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ n ∈ ( m,M ) a n ( f ) ϕ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( I,ω ) ≤ C ( p ) M γ ( p ′ ) ε ( M − m ) k f k L p ( I,ω ) . (17)5 roof. By orthogonality and Holder’s inequality, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ n ∈ ( m,M ) a n ( f ) ϕ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( I,ω ) = X λ n ∈ ( m,M ) a n ( f ) ≤ X λ n ∈ ( m,M ) k ϕ n k L p ′ ( I,ω ) k f k L p ( I,ω ) From condition ( A ), we have k ϕ n k L p ′ ( I,ω ) ≤ n γ ( p ′ ) . Also,by using condition ( D ), we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ n ∈ ( m,M ) a n ( f ) ϕ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( I,ω ) ≤ X λ n ∈ ( m,M ) n γ ( p ′ ) k f k L p ( I,ω ) ≤ C X λ n ∈ ( m,M ) λ γ ( p ′ ) ε n k f k L p ( I,ω ) ≤ CM γ ( p ′ ) ε X λ n ∈ ( m,M ) k f k L p ( I,ω ) ≤ CM ( γ ( p ′ ) ε ) ( M − m ) k f k L p ( I,ω ) . Then we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X λ n ∈ ( m,M ) a n ( f ) ϕ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( I,ω ) ≤ CM ( γ ( p ′ ) ε ) ( M − m ) k f k L p ( I,ω ) . Proof of Theorem1. We should mention here that some parts of the proof of this theorem are inspiredfrom [6]. Without loss of generality, we can consider 1 ≤ p < δR . In order to do so, let φ ∈ C ∞ ( R )with support on (1 / , 2) such that X k ∈ Z φ (2 k t ) = 1 and φ ( t ) = 1 − + ∞ X k =1 φ (2 k t ) for all t > 0. We define φ δR,k ( t ) = (cid:18) − tR (cid:19) δ + φ (cid:18) k (1 − tR ) (cid:19) . We recall that, from [6], this last function has the following properties :1. supp (cid:16) φ δR,k (cid:17) ⊆ ( R (1 − − k +1 ) , R (1 − − k − )),2. sup t ∈ R | φ δR,k ( t ) | ≤ C − kδ ,3. ∀ N ≥ , there exists C N > | ∂ Nt φ δR,k ( t ) | ≤ C N (cid:16) k R (cid:17) N . Furthermore, we denote byΨ δR,k .f = ∞ X n =0 φ δR,k ( λ n ) a n ( f ) ϕ n k = 1 , , · · · (18)6hen, we haveΨ δR f = ∞ X n =0 (cid:18) − λ n R (cid:19) δ + a n ( f ) ϕ n = ∞ X n =0 φ (1 − λ n R ) (cid:18) − λ n R (cid:19) δ + a n ( f ) ϕ n + ∞ X n =0 + ∞ X k =1 φ (2 k (1 − λ n R )) ! (cid:18) − λ n R (cid:19) δ + a n ( f ) ϕ n = ∞ X n =0 φ (1 − λ n R ) (cid:18) − λ n R (cid:19) δ + a n ( f ) ϕ n + [ log( R )log(2) ] X k =1 ∞ X n =0 φ δR,k ( λ n ) a n ( f ) ϕ n + ∞ X k = [ log( R )log(2) ] +1 ∞ X n =0 φ δR,k ( λ n ) a n ( f ) ϕ n = ψ δR, f + [ log( R )log(2) ] X k =1 Ψ δR,k f + R δR f. It is clear that the main term is the second one. With the same approach used in [6], we willprove the following proposition : Proposition 1. Let ≤ p < and δ > δ ( p ) = γ ( p ′ ) ε . There exists β > such that for every f ∈ L p ( I, w ) , we have (cid:13)(cid:13) Ψ δR,k f (cid:13)(cid:13) L p ( I,w ) ≤ C − kβ k f k L p ( I,w ) , (19) where C is a constant independent of R and f .Proof. Let x = a + b ∈ ( a, b ) and r = b − a > x − r, x + r ) ⊆ ( a, b ) . Note that, forevery 1 ≤ k ≤ h log( R )log(2) i = k R , we have r αk = (cid:16) k R (cid:17) µ ( p ) r < r where µ ( p ) = ( γ ( p ′ ) ε + )( p − ) . So we noticethat I = ( a, b ) = ( x − r αk , x + r αk ) ∪ { y ∈ ( a, b ) , | y − x | > r αk } = I αk, ∪ I αk, .We start by providing an L p bound of (cid:13)(cid:13)(cid:13) Ψ δR,k (cid:13)(cid:13)(cid:13) L p ( I αk, ,ω ) . To do so, we proceed in the way toreduce the L p inequality (19) to certain ( L p , L ) inequality using the last lemma.Using Parseval formula and the fact that supp (cid:16) φ δR,k (cid:17) ⊆ ( R k, , R k, ), where R k, = R (1 − − k +1 )and R k, = R (1 − − k − ) , we have (cid:13)(cid:13) Ψ δR,k f (cid:13)(cid:13) L ( I,w ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =0 φ δR,k ( λ n ) a n ( f ) ϕ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( I,w ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X R k, ≤ λ n ≤ R k, φ δR,k ( λ n ) a n ( f ) ϕ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( I,w ) , Using the previous lemma with m = R k, , M = R k, and the fact that sup t ∈ R | φ δR,k ( t ) | ≤ C − kδ , onegets (cid:13)(cid:13) Ψ δR,k f (cid:13)(cid:13) L ( I,w ) ≤ C − kδ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X R k, ≤ λ n ≤ R k, a n ( f ) ϕ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( I,w ) ≤ C − kδ R (2 γ ( p ′ ) ε ) (cid:18) R k +1 (cid:19) k f k L p ( I,w ) . (cid:13)(cid:13) Ψ δR,k f (cid:13)(cid:13) L ( I,w ) ≤ C − k ( δ + ) R ( γ ( p ′ ) ε )+ ) k f k L p ( I,w ) (20)By combining H¨older inequality and (20), we obtain (cid:13)(cid:13) Ψ δR,k f (cid:13)(cid:13) L p ( I αk, ,w ) ≤ ( µ ( I k, )) p − (cid:13)(cid:13) Ψ δR,k f (cid:13)(cid:13) L ( I αk, ,w ) ≤ (2 r αk ) p − (cid:13)(cid:13) Ψ δR,k f (cid:13)(cid:13) L ( I,w ) ≤ C − k ( δ − γ ( p ′ ) ε ) k f k L p ( I,w ) . (21)Let s δR,k ( u, v ) = ∞ X n =0 φ δR,k ( λ n ) ϕ n ( x ) ϕ n ( y ) be the kernel of Ψ δR,k . We just have to find an estimateof || Ψ δR,k f || L p ( I αk, ,w ) , so we will use the Schur test with the symmetric property of s δR,k , then itsuffices to prove the following inequalitysup u ∈ I αk, (cid:13)(cid:13) s δR,k ( u, . ) (cid:13)(cid:13) L ( I αk, ) ≤ C − kε for some ε > C > p. We consider g δR,k ( λ ) = (cid:16) − λ R (cid:17) δ + e λ /R φ (2 k (1 − λ R )) satisfying the following properties, see [6]1. For every non-negative integer i there exists a constant C i such that for all s > Z | t |≥ s | ˆ g δR,k ( t ) | dt ≤ C i s − i R − i/ ( i − δ ) k (22)2. (cid:13)(cid:13)(cid:13) g δR,k ( √L ) (cid:13)(cid:13)(cid:13) ( L ,L ) ≤ C − kδ . (23)For our purpose, we will consider such a positive self-adjoint operator L on L ( R ) such that thesemigroup e − t L , generated by −L , has the kernel p t ( x, y ) obeying the Gaussian upper bound | p t ( u, v ) | ≤ C √ t exp (cid:18) − | u − v | Ct (cid:19) . (24)for a constant C > 0. (see [12])For all u ∈ R and t > 0, one gets the following estimate k p t ( u, . ) k L ( R ) ≤ C. (25)On the other hand, there exists i ∈ N such that 2 i o − < R µ ( p ) < i and we can see that I αk, ⊆ ∪ µ ( p ) k − i ≤ j ≤ D j where D j = { y, j r ≤ | y − x | < j +1 r } . Since, L is a positive self-adjoint operator, then it’s clearthat φ δR,k ( L ) = g δR,k ( √L ) exp ( −L /R ) . (26)Hence one gets s δR,k ( u, v ) = g δR,k ( √L ) (cid:0) p /R ( u, . ) (cid:1) ( v )= g δR,k ( √L ) (cid:0) p /R ( u, . ) χ { w, | x − w | < j − r } (cid:1) ( v ) + g δR,k ( √L ) (cid:0) p /R ( u, . ) χ { w, | x − w |≥ j − r } (cid:1) ( v )= s δ, R,k ( u, v ) + s δ, R,k ( u, v ) . g δR,k is an even function, with the inversion formula, we have g δR,k ( √ λ ) = 1 √ π Z R ˆ g δR,k ( t ) cos ( t √ λ ) dt. Hence, we obtain s δ, R,k ( u, v ) = 1 √ π Z R ˆ g δR,k ( t ) cos ( t √L ) (cid:0) p /R ( u, . ) χ { w, | x − w | < j − r } (cid:1) ( v ) dt. Moreover, the operator cos ( t √L ) is bounded in L with support kernel K t satisfying, see [12, 23]Supp ( K t ) = { ( u, v ) ∈ R , | u − v | ≤ c | t |} From (23), (25) and the previous analysis, one gets (cid:13)(cid:13)(cid:13) s δ, R,k ( u, . ) (cid:13)(cid:13)(cid:13) L ( D j ) = 1 √ π (cid:13)(cid:13)(cid:13)(cid:13)Z R ˆ g δR,k ( t ) cos ( t √L ) (cid:0) p /R ( u, . ) χ { w, | x − w | < j − r } (cid:1) dt (cid:13)(cid:13)(cid:13)(cid:13) L ( D j ) = 1 √ π (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z | t | > j − rc ˆ g δR,k ( t ) cos ( t √L ) (cid:0) p /R ( u, . ) χ { w, | x − w | < j − r } (cid:1) dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( D j ) ≤ µ / ( D j ) √ π Z | t | > j − rc | ˆ g δR,k ( t ) | (cid:13)(cid:13) p /R ( u, . ) (cid:13)(cid:13) L ( D j ) dt ≤ C √ π j +12 Z | t | > j − rc | ˆ g δR,k ( t ) | dt Let i > µ + µ +1 − p ) where µ = γ ( p ′ ) ε > . Then by (22), there exists a constant C i > (cid:13)(cid:13)(cid:13) s δ, R,k ( u, . ) (cid:13)(cid:13)(cid:13) L ( D j ) ≤ C i √ π j/ ( 2 j c ) − i R − i/ ( i − δ ) k ≤ C i √ π c i ( i − δ ) k j (1 / − i ) . Then, we obtain (cid:13)(cid:13)(cid:13) s δ, R,k ( u, . ) (cid:13)(cid:13)(cid:13) L ( I k, ) ≤ X µ ( p ) k − i ≤ j ≤ (cid:13)(cid:13)(cid:13) s δ, R,k ( u, . ) (cid:13)(cid:13)(cid:13) L ( D j ) ≤ C i √ π c i ( i − δ ) k X µ ( p ) k − i ≤ j ≤ j (1 / − i ) ≤ C i √ π c i ( i − δ ) k ( i − / i − µ ( p ) k +1) ≤ C ′ i − kε . From our assumption on i, ε = δ − i + ( i − / µ + ( p − ) )) > . Then, to have an estimate of the kernel s δ, R,k on L ( I δk, ), it suffices to find an estimate of the kernel s δ, R,k on L ( I δk, ). 9rom (23), (24) and using the fact that R ≤ R µ ( p ) , one gets the following inequality (cid:13)(cid:13)(cid:13) s δ, R,k ( u, . ) (cid:13)(cid:13)(cid:13) L ( D j ) = Z D j | g δR,k ( √L ) (cid:0) p /R ( u, . ) χ { w, | w − x | > j − r } (cid:1) ( v ) | dv ≤ (cid:13)(cid:13)(cid:13) g δR,k ( √L ) (cid:13)(cid:13)(cid:13) ( L ,L ) (cid:13)(cid:13) p /R ( u, . ) χ { w, | w − x | > j − r } (cid:13)(cid:13) L ( D j ) ≤ C − kδ √ Re ( − CR j − ) ( µ ( D j )) / ≤ C − kδ i j e − C i j ) . Hence, we conclude that (cid:13)(cid:13)(cid:13) s δ, R,k ( u, . ) (cid:13)(cid:13)(cid:13) L ( I k, ) ≤ X µ ( p ) k − i ≤ j ≤ (cid:13)(cid:13)(cid:13) s δ, R,k ( u, . ) (cid:13)(cid:13)(cid:13) L ( D j ) ≤ C − kδ X i = i o + j ≥ µ ( p ) k i e − C i ≤ C ′ − kδ . Proposition 2. Let ≤ p ≤ and δ > δ ( p ) = γ ( p ′ ) ε , then for all f ∈ L p ( I, w ) , we have (cid:13)(cid:13) ψ δR, f (cid:13)(cid:13) L p ( I,w ) ≤ C k f k 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 estimateof (cid:13)(cid:13) ψ δR, f (cid:13)(cid:13) L p ( I ,w ) and (cid:13)(cid:13) ψ δR, f (cid:13)(cid:13) L p ( I ,w ) for all f ∈ L p ( I, w ) , where I = ( a, b ) = I ∪ I with I =( x − r α , x + r α ) and I = { y, | y − x | > r α } where r α = rR µ ( p ) . To conclude the theorem’s proof it suffices to find a uniform bound of R δR . Proposition 3. Let ≤ p ≤ and δ > δ ( p ) = γ ( p ′ ) ε , then for all f ∈ L p ( I, w ) , we have (cid:13)(cid:13) R δR f (cid:13)(cid:13) L p ( I,w ) ≤ C k f k L p ( I,w ) . (28) where C depends only on p .Proof. From Holder’s inequality and the previous lemma, we have (cid:13)(cid:13) R δR f (cid:13)(cid:13) L p ( I,w ) ≤ p − ) (cid:13)(cid:13) R δR f (cid:13)(cid:13) L ( I,w ) ≤ p − ) ∞ X k = K R +1 ∞ X n =0 (cid:13)(cid:13) φ δR,k ( λ n ) a n ( f ) ϕ n (cid:13)(cid:13) L ( I,w ) ≤ C p − ) ∞ X k = K R +1 − kδ X R k, ≤ λ n ≤ R k, k a n ( f ) ϕ n k L ( I,w ) ≤ C p − ) ∞ X k = K R +1 − k ( δ + ) R + γ ( p ′ ) ε ) k f k L p ( I,ω ) ≤ C p − ) − δ + ) (cid:0) [ log( R )log(2) ] +1 (cid:1) R + γ ( p ′ ) ε ) k f k L p ( I,ω ) ≤ C p − ) R − δ − ( γ ( p ′ ) ε )) k f k L p ( I,ω ) (cid:13)(cid:13) R δR f (cid:13)(cid:13) L p ( I,w α,β ) ≤ C ( p − ) R − ( δ − ( γ ( p ′ ) ε )) k f k L p ( I,ω α,β ) ≤ C ( p ) k f k L p ( I,ω α,β ) . Corollary 1. Under the notation and conditions of the previous Theorem, we have for all f ∈ L p ( I, w ) Ψ δR f → f as R → ∞ . (29) Proof. Step1: We prove that, for every f ∈ C ∞ ( I, R ), Ψ δR f → f in L p ( I, ω ). Note that (cid:12)(cid:12)(cid:12)(cid:16) − λ n R (cid:17) δ + h f, ϕ n i L ( I,ω ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) h f, ϕ n i L ( I,ω ) (cid:12)(cid:12)(cid:12) = 1 λ n (cid:12)(cid:12)(cid:12) h f, L ϕ n i L ( I,ω ) (cid:12)(cid:12)(cid:12) = 1 λ n (cid:12)(cid:12)(cid:12) hL f, ϕ n i L ( I,ω ) (cid:12)(cid:12)(cid:12) = · · · = 1 λ kn (cid:12)(cid:12)(cid:12)(cid:10) L k .f, ϕ n (cid:11) L ( I,ω ) (cid:12)(cid:12)(cid:12) ≤ n − kε (cid:13)(cid:13) L k .f (cid:13)(cid:13) L ( I,ω ) (30)Since k ϕ n k L ( I,ω ) ≤ n γ ( p ) , it suffices to take k big enough to have γ ( p ) − kε < − L p ( I, ω ).As (cid:13)(cid:13) Ψ δR .f − f (cid:13)(cid:13) = ∞ X n =0 (cid:16) (1 − λ n R ) δ + − (cid:17) | a n ( f ) | → R → ∞ , the result remains true for1 ≤ p < ∞ . Step2: For all ε > 0. By density of C ∞ ( I, R ) in L p ( I, ω ), there exists g ∈ C ∞ ( I, R ) such that k f − g k L p ( I,ω ) < ε and there exists R > (cid:13)(cid:13) Ψ δR .f − Ψ δR .g (cid:13)(cid:13) L p ( I,ω ) < ε .By writing, (cid:13)(cid:13) Ψ δR .f − f (cid:13)(cid:13) L p ( I,ω ) ≤ (cid:13)(cid:13) Ψ δR .f − Ψ δR .g (cid:13)(cid:13) L p ( I,ω ) + (cid:13)(cid:13) Ψ δR .g − g (cid:13)(cid:13) L p ( I,ω ) + k f − g k L p ( I,ω ) , one gets the desired result. The ail of this section is apply the main result of the study of L p -Bochner Riesz convergence to thefamily of Jacobi polynomials. In addition we will further give a necessary condition. More precisely,we give the following result : Theorem 2. Let α ≥ β > − / , c > and ( e P ( α,β ) k ) n ≥ be the family of normalized Jacobi polyno-mials. For a smooth function f on I = ( − , , define Ψ δR f = ∞ X n =0 (cid:16) − χ n R (cid:17) δ + D f, e P ( α,β ) k E ω α,β e P ( α,β ) k . Then, for every ≤ p < ∞ such that p = (2 − α +3 / ) , Ψ δR extends to a bounded operator L p ( I, ω α,β ) → L p ( I, ω α,β ) . Further Ψ δR f → f for every f ∈ L p ( I, ω α,β ) if and only if > max { γ ( p ′ )2 , } , where p ′ is the conjugate of p and γ ( p ) = if < p < p ′ ǫ if p = p ′ α + 1) h − p i − if p > p ′ α + if p = 1 , ∞ . Jacobi polynomials (Sufficient condition) Let I = ( − , ω α,β ( x ) = (1 − x α ) α (1 + x ) β , α, β > − / ≤ p < ∞ , the associated L p norms are given by k f k L p ( I,ω α,β ) = (cid:18)Z − | f ( x ) | p (1 − x ) α (1 + x ) β d x (cid:19) p . The Jacobi polynomials are the orthonormal family of polynomials with respect to the inner productassociated with k·k L ( I,ω α,β ) with leading coefficient being non-negative.Alternatively, we define the (non-normalized) Jacobi polynomials P ( α,β ) k through the boundedeigenfunctions associated with χ n = n ( n + α + β + 1) of the following Sturm-Liouville operator D = 1 ω α,β ( x ) ddx (cid:16) ω α +1 ,β +1 ( x ) ddx (cid:17) We consider the normalized Jacobi polynomials e P ( α,β ) k = (cid:13)(cid:13)(cid:13) P ( α,β ) k (cid:13)(cid:13)(cid:13) − L ( I,ω α,β ) P ( α,β ) k which form anorthonormal basis of L ( I, ω α,β ).Moreover, let p = 2 − α +3 / and p ′ = 2 + 1 α + 1 / < p < ∞ , the L p -norm ofJacobi polynomials is given by Aptekarev, Buyarov and Degeza [2] : k e P ( α,β ) n k L p ( I,ω α ) = C ( α, p ) + ◦ (1) if 1 < p < p ′ C ( α, p ) log /p ( n )(1 + ◦ (1)) when p = p ′ n ( α +1 / − p ′ p ) when p > p ′ (31)with C ( α, p ) a generic constant depending only on α and p . Moreover, from [24], we have k e P ( α,β ) n k ∞ ≤ Cn α + , that is k e P ( α,β ) n k L ( I,ω α ) ≤ Cn α + . One concludes that Jacobi polynomials satisfy k e P ( α,β ) n k L p ( I,ω α ) ≤ Cn γ ( p ) (32)where γ ( p ) = < p < p ′ ǫ if p = p ′ α + 1) h − p i − if p > p ′ α + if p = 1 for any ǫ > . Finally, we can see that the Jacobi polynomials satisfy condition [(A)].It is well known that the corresponding Liouville operator have χ n = n ( n + α + β +1) as the ( n +1)-theigenvalue then we have χ n ≥ n . Moreover, straightforward computation for a fixed 0 ≤ m < M such that M − m > X χ n ∈ ( m,M ) X n ∈ (cid:16) ( m +( α + β +12 ) ) − ( α + β +12 ) , ( M +( α + β +12 ) ) − ( α + β +12 ) (cid:17) ≤ C ( M − m )12o, we can see that condition (B) is also satisfied. Jacobi polynomials (necessary condition) For the necessary condition, we need to use the transferring theorem from the uniform bounded-ness of Ψ δR to the uniform boundedness of the Hankel multiplier transform operator M α defined by M α ( f ) = H α ( m H α ( f )) where m ( x ) = (cid:0) − x (cid:1) δ + and H α is the modified hankel operator defined by H α ( f )( x ) = R ∞ J α ( xy )( xy ) α f ( y ) y α +1 dy and that is possible from [14]. Moreover, from [10], the uniformboundedness of the last is hold true if and only if δ > max { α + 1) | p − | − , } . It’s easy to checkthat max { α + 1) | p − | − , } ≥ max { γ ( p ′ )2 , } for every p = 2 − α +3 / , then we get our necessarycondition. We apply now our main result to the case of the weighted and Hankel prolate. More precisely : Theorem 3. Let α > − / , δ and c be two positive number and ( ψ ( α ) n,c ) n ≥ be the family of weightedprolate spheroidal wave functions. For a smooth function f on I = ( − , , we define Ψ δR f = ∞ X n =0 (cid:18) − χ n,α ( c ) R (cid:19) δ + D f, ψ ( α ) n,c E L ( I,ω α ) ψ ( α ) n,c . Then, for every ≤ p < ∞ , Ψ δR extends to a bounded operator L p ( I, ω α ) → L p ( I, ω α ) . Further, Ψ δR f → f for every f ∈ L p ( I, ω α ) such that δ > max { γ ( p ′ )2 , } and p = 2 − α +3 / where γ ( p ) = if < p < p ′ ǫ if p = p ′ α + 1) h − p i − if p > p ′ α + 1 if p = 1 . Proof. We recall that from (13), the GPSWFs are the eigenfunctions of the Sturm-Liouville oper-ator L ( α ) c . Also, note that the ( n + 1) − th eigenvalue χ αn ( c ) of L ( α ) c satisfies the following classicalinequalities, n ≤ n ( n + 2 α + 1) ≤ χ αn ( c ) ≤ n ( n + 2 α + 1) + c , ∀ n ≥ . Moreover, for every 0 ≤ m < M such that M − m > 1, we have X χ n,α ( c ) ∈ ( m,M ) ≤ X n ( n +2 α +1) ∈ (max(0 ,m − c ) ,M ) ≤ X ( n + α +1 / − ( α +1 / ∈ (max(0 ,m − c ) ,M ) ≤ X n ∈ (cid:16) (max(0 ,m − c )+( α +1 / ) − (1 / α ) , ( M +( α +1 / ) − (1 / α ) (cid:17) ≤ C ( M − m )It follows that condition (B) is satisfied.Form [4] Lemma 2 . 6, one can conclude that condition (A) is satisfied for weighted prolate spheroidalwave functions for 1 < p < ∞ . Moreover, it has been shown in [15] that (cid:13)(cid:13)(cid:13) ψ ( α ) n,c (cid:13)(cid:13)(cid:13) ∞ ≤ C (cid:16) χ αn ( c ) (cid:17) α +12 . Then by using (14) we obtain, (cid:13)(cid:13)(cid:13) ψ ( α ) n,c (cid:13)(cid:13)(cid:13) ≤ C (cid:16) χ αn ( c ) (cid:17) α +12 ≤ Cn α +1 . heorem 4. Let α > − / and c > and ( ϕ ( α ) n,c ) n ≥ be the family of Hankel prolate spheroidalwave functions. For a smooth function f on I = (0 , , define Ψ δR f = ∞ X n =0 (cid:18) − χ n,α ( c ) R (cid:19) δ + D f, ϕ ( α ) n,c E L (0 , ϕ ( α ) n,c . Then, for every ≤ p < ∞ , Ψ δR extends to a bounded operator L p (0 , → L p (0 , . Further Ψ δR f → f in L p (0 , for every f ∈ L p (0 , such that δ > max { γ ( p ′ )2 , } where γ ( p ) = p − if < p < ǫ − if p = 4 h p − i if p > if p = 1 . .Proof. From (9), the CPSWFs basis ϕ αn,c is the n − th order bounded eigenfunction of the positiveself-adjoint operator L αc associated with the eigenvalue χ n,α ( c ) . The 2d-Slepian family ϕ αn,c form anorthonormal basis of L (0 , 1) and the eigenvalues family χ n,α ( c ) satisfy the following inequality4 n ≤ (2 n + α + 1 / n + α + 3 / ≤ χ n,α ( c ) ≤ (2 n + α + 1 / n + α + 3 / 2) + c Moreover, for every 0 ≤ m < M such that M − m > 1, we have X χ n,α ( c ) ∈ ( m,M ) ≤ X (2 n + α +1 / n + α +3 / ∈ (max(0 ,m − c ) ,M ) ≤ X (2 n + α +1) − / ∈ (max(0 ,m − c ) ,M ) ≤ X n ∈ (cid:16) (max(0 ,m − c )+1 / − (1+ α ) , ( M +1 / − (1+ α ) (cid:17) ≤ C ( M − m )we can now see that condition (B) is satisfied.Form [4] Lemma 2 . < p < ∞ . 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