Discrete Prolate Spheroidal Wave Functions: Further spectral analysis and some related applications
DDiscrete Prolate Spheroidal Wave Functions: Further spectral analysisand some related applications.
Mourad Boulsane a , NourElHouda Bourguiba a and Abderrazek Karoui a a University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerte, Jarzouna 7021,Tunisia.
Abstract — For fixed W ∈ (cid:0) , (cid:1) and positive integer N ≥ , the discrete prolate spheroidalwave functions (DPSWFs), denoted by U Nk,W , ≤ k ≤ N − (cid:101) Q N,W , defined on L ( − / , / , with kernel K N ( x, y ) = sin( Nπ ( x − y ))sin( π ( x − y )) [ − W,W ] ( y ) . It is well known that the DPSWF’s have a wide range of classi-cal as well as recent signal processing applications. These applications rely heavily on the propertiesof the DPSWFs as well as the behaviour of their eigenvalues (cid:101) λ k,N ( W ) . In his pioneer work [17], D.Slepian has given the properties of the DPSWFs, their asymptotic approximations as well as theasymptotic behaviour and asymptotic decay rate of these eigenvalues. In this work, we give furtherproperties as well as new non-asymptotic decay rates of the spectrum of the operator (cid:101) Q N,W . Inparticular, we show that each eigenvalue (cid:101) λ k,N ( W ) is up to a small constant bounded above by thecorresponding eigenvalue, associated with the classical prolate spheroidal wave functions (PSWFs).Then, based on the well established results concerning the distribution and the decay rates of theeigenvalues associated with the PSWFs, we extend these results to the eigenvalues (cid:101) λ k,N ( W ). Also,we show that the DPSWFs can be used for the approximation of classical band-limited functionsand they are well adapted for the approximation of functions from periodic Sobolev spaces. Finally,we provide the reader with some numerical examples that illustrate the different results of this work.2010 Mathematics Subject Classification. Primary 42A38, 15B52. Secondary 60F10, 60B20. Key words and phrases.
Band-limited sequences, eigenvalues and eigenfunctions, discrete prolatespheroidal wave functions and sequences, eigenvalues distribution and decay rate.
A breakthrough in the theory and the construction of the discrete prolate spheroidal wave functionsis due to D. Slepian [17], who has studied most of the properties, the numerical computations, aswell as the asymptotic behaviours of the DPSWFs and their associated eigenvalues. Note that forfixed W ∈ (cid:0) , (cid:1) and integers N ∈ N , N ≥
1, the DPSWF’s are characterized as the amplitudespectra (Fourier series) of index-limited complex sequences with index support [[ N , N + N − { N , . . . , N + N − } , that are most concentrated in the interval ( − W, W ) . For the sake of simplicityof the notations an without loss of generality, we will only consider the N = 0 in this work. As it willdescribed later on, the DPSWFs’s are closely related to their associated Discrete Prolate SpheroidalSequences (DPSS’s). These DPSS’s are infinite sequences in (cid:96) ( C ) with amplitude spectra supportedin [ − W, W ] and with coefficients most concentrated in the index range [[0 , . . . , N − . The DPSS’ssequences have been successfully used in various classical as well as fairly recent applications fromthe signal processing area. To cite but a few, the prediction of white noise random samples of discrete Corresponding author: Abderrazek Karoui, Email: [email protected] work was supported by the Tunisian DGRST research grant UR 13ES47. a r X i v : . [ m a t h . C A ] M a y ignals with bandwidth W , [17], the DPSS’s based scheme for compressive sensing [7], parametricwaveform and detection of extended targets [21] and fast algorithms for Fourier extension [1], etc.It has been shown in [17], that the solution of the energy maximization problem, associatedwith the DPSWF’s is given by the first eigenfunction corresponding to the largest eigenvalue of thefollowing eigenproblem (cid:90) W − W sin( N π ( y − x ))sin( π ( y − x )) h ( y ) dy = λh ( x ) , x ∈ ( − / , / . (1)Therefore, the different DPSWF’s U Nk,W are the N eigenfunctions of a finite rank integral operator (cid:101) Q W,N , that is (cid:101) Q W,N ( U Nk,W )( x ) = (cid:90) W − W sin( N π ( x − y ))sin( π ( x − y )) U Nk,W ( y ) dy = (cid:101) λ k,N ( W ) U Nk,W ( x ) . (2)Here, 1 > (cid:101) λ ,N ( W ) > (cid:101) λ ,N ( W ) > · · · > (cid:101) λ N − ,N ( W ) is the sequence of the associated eigenval-ues, arranged in the decreasing order. The N DPSWF’s form an orthonormal system of both L ( − W, W ) , W ∈ (cid:0) , (cid:1) and L ( − / , / . More precisely, they satisfy the following double or-thogonality properties (cid:90) W − W U Nk,W ( x ) U Nj,W ( x ) dx = (cid:101) λ k,N ( W ) δ k,j , (cid:90) / − / U Nk,W ( x ) U Nj,W ( x ) dx = δ k,j , k, j = 0 , . . . , N − . (3)From [17], the DPSWFs are related to the DPSS’s by the following rule. Let V Nk,W = ( v ( k )0 , ....., v ( k ) N − ) T ,k = 0 , .., N − N vectors obtained by truncating the DPSS’s to the index set [[0 , N − . Then, these truncated DPSS’s are the N eigenvectors of the Toeplitz matrix ρ N,W = (cid:20) sin(2 π ( n − m ) W ) π ( n − m ) (cid:21) n,m =0 ,..,N − . (4)Moreover, we have U Nk,W ( x ) = (cid:15) kN − (cid:88) n =0 v ( k ) n e − iπ ( N − − n ) x , (cid:15) k = (cid:26) , k even; i, k odd. (5)Note that the matrix ρ N,W has the same spectrum as the integral operator (cid:101) Q W,N , that is the DP-SWFs U Nk,W and the corresponding truncated DPSS’s V Nk,W are associated with same eigenvalues (cid:101) λ k,N ( W ) . Also, it is interesting to note that the spectrum associated with the DPSWFs has somesurprising similarities with the spectrum associated with the classical PSWFs, that have been intro-duced and greatly investigated since the early 1960’s, by D. Slepian and his co-authors H. Landauand H. Pollak, see [10, 16, 17]. We recall that for a given real number c >
0, called the bandwidth,the PSWFs ( ψ n,c ( · )) n ≥ constitute an orthonormal basis of L ([ − , +1]) , an orthogonal system of L ( R ) and an orthogonal basis of the Paley-Wiener space B c , given by B c = (cid:110) f ∈ L ( R ) , Support (cid:98) f ⊂ [ − c, c ] (cid:111) . (6)Here, (cid:98) f denotes the Fourier transform of f ∈ L ( R ) . They are eigenfunctions of the Sinc-kerneloperator defined on L ([ − , , that is Q c ( ψ n,c )( x ) = (cid:90) − sin c ( x − y ) π ( x − y ) ψ n,c ( y ) dy = λ n ( c ) ψ n,c ( x ) , x ∈ [ − , . (7)2nlike the classical case, where there exist a rich literature on the behaviour and the decay rates(both asymptotic and non-asymptotic) of the eigenvalues λ n ( c ) , see for example [4, 6, 9, 10, 16, 18],the counterpart literature for the (cid:101) λ k,N ( W ) is still very limited. The main existing decay rate resultfor the (cid:101) λ k,N ( W ) is an asymptotic one and it goes back to [17], where it has been shown that forfixed W ∈ (cid:0) , (cid:1) and ε ∈ (cid:0) , W − , we have (cid:101) λ k,N ( W ) ≤ C ( W, ε ) e − C ( W,ε ) N , ∀ k ≥ (cid:100) N W (1 + ε ) (cid:101) , N ≥ N ( W, ε ) , (8)for some constants C ( W, ε ) , C ( W, ε ) and N ( W, ε ) ∈ N that depend on W, ε.
The previous estimateis asymptotic and the dependence of the previous constants does not have explicit estimates. Theprevious decay rate has been recently generalized in [22] to the multiband DPSS’s setting. Moreover,in this last reference and by using some advanced matrix analysis and computations techniques, theauthors have given the following distribution of the (cid:101) λ k,N ( W ) . If W is an union of J pairwise disjointintervals with W ⊂ (cid:0) − , (cid:1) and ε ∈ (cid:0) , (cid:1) , then { k : ε ≤ (cid:101) λ k,N ( W ) ≤ − ε } ≤ J π log( N −
1) + π N − N − ε (1 − ε ) . (9)In this work, for c = πN W and by comparing the Hilbert-Schmidt norms (cid:107) (cid:101) Q W,N (cid:107) HS and (cid:107)Q c (cid:107) HS ofthe integral operators (cid:101) Q W,N and Q c , given by (2) and (7), we prove that for J = 1 , we have { k : ε ≤ (cid:101) λ k,N ( W ) ≤ − ε } ≤ π log(2 N W ) + 0 . − W + W c sin (2 c ) ε (1 − ε ) , c = πN W. (10)It can be easily checked that for J = 1 , N ≥ πN W ≥ , our estimate (10) improves theestimate (9). Also, the comparison of the previous Hilbert-Schmidt norms, together with the use ofthe Wielandt-Hoffman inequality, we prove that for sufficiently small W, the spectrum of (cid:101) Q W,N iswell approximated in the (cid:96) -norm by the spectrum of the Sinc-kernel operator Q c , c = πN W. Moreprecisely, for any N ≥ W ∈ (cid:0) , (cid:1) , we have (cid:32) N − (cid:88) k =0 (cid:12)(cid:12)(cid:12)(cid:101) λ k,N ( W ) − λ k ( c ) (cid:12)(cid:12)(cid:12) (cid:33) ≤ W (cid:18) π W π ) (cid:19) , c = πN W. (11)Also by taking advantage from a connection between the energy maximization problems associatedto the DPSWFs and the classical PSWFs, ψ n,c with c = πN W, we prove the following unexpectedand important result relating the (cid:101) λ Nn,W and the λ n ( c ) , (cid:101) λ n,N ( W ) ≤ A W λ n ( c ) , ≤ n ≤ N − , (12)where π ≤ A W = 2 π cos ( πW ) (cid:18) − W (cid:19) ≤ . (13)Thanks to the estimate (12), all the existing known asymptotic and non-asymptotic decay ratesfor the λ n ( c ) are transmitted to the (cid:101) λ n,N ( W ) . For example, based on the recent non-asymptoticestimates of the λ n ( c ) , given in [6], one concludes that under the condition that for sufficiently awayfrom the plunge region of the spectrum, that is for 2 ≤ eπ N W ≤ n ≤ N − , we have (cid:101) λ n,N ( W ) ≤ e − (2 n +1) log (cid:0) n +2 eπNW (cid:1) . (14)Moreover, for n close to the plunge region around [2 N W ] , there exists a constant η > , such that (cid:101) λ n,N ( W ) ≤ e − η n − NW log( πNW )+5 , N W + log( πN W ) + 6 ≤ n ≤ πN W. (15)3s applications of the DPSWF’s that we consider in this work, we first get an estimate of theunknown constant appearing in the Tur`an-Nazarov concentration inequality. Then, we check thatthere exists N ≥ [2 N W ] − , such that the eigen-space spanned by the first N dilated DPSWFs √ W U
Nk,W ( W · ) is approximated by the eigen-space spanned by the corresponding classical ψ k,c . Also,we check that these DPSWF’s are well adapted for the spectral approximation of functions from theperiodic Sobolev space H sper ( − / , / , s > . Finally, this work is organized as follows. In section 2, we give some mathematical preliminariesrelated to the properties and the computations of DPSWFs and DPSS’s and their associated eigen-values. Moreover, we give some first estimates for the eigenvalues associated with the DPSWF’s.In section 3, we study some interesting connections between the DPSWFs and their correspondingclassical ψ n,c , c = πN W. Based on these connections, we deduce various results on the distributionand the decay rates of the eigenvalues (cid:101) λ n,N ( W ) . Section 4 is devoted to the previous proposed ap-plications of the DPSWF’s. In the last section 5, we give some numerical examples that illustratethe different results of this work.
In this paragraph, we first recall from the literature, some properties and computational methods forthe DPSWFs and their associated eigenvalues. Also, we give some first estimates of the eigenvaluesassociated with the DPSWFs. These estimates are obtained in a fairly easy way by using the Min-Max characterization of the eigenvalues of self-adjoint compact operators. More involved and preciseestimates of the eigenvalues (cid:101) λ n,N ( W ) , is the subject of the next section 3.We recall from [17], that the DPSWFs ( U Nn,W ) ≤ n ≤ N − are the eigenfunctions of the positive, self-adjoint finite rank integral operator (cid:101) Q W,N , given by (2). This last eigen-problem is a consequence ofthe fact that the DPSWF’s. Among the space S N of all sequences xxx = ( x n ) n ∈ l ( C ) with elementsindexed on [[0 , N ]] , so that their amplitude spectra (cid:98) xxx ( t ) = N (cid:88) k =0 x k e iπkt , find those sequences withamplitude spectra most concentrated on ( − W, W ) , that is solve the maximization problem U = arg max xxx ∈ S N (cid:107) (cid:98) xxx ( t ) (cid:107) L ( − W,W ) (cid:107) (cid:98) xxx ( t ) (cid:107) L ( − / , / . (16)Note that the DPSWFs ( U Nn,W ) are periodic. They have period 1 if N is odd and period 2 if N iseven. In either case we have U Nk,W ( x + 1) = ( − N − U Nk,W ( x ) , x ∈ [ − / , / . Also, the associated eigenvalues satisfy the following relation, (cid:101) λ k,N (cid:16) − W (cid:17) = 1 − (cid:101) λ N − k − ,N ( W ) , ∀ k = 0 , . . . , N − . (17)Moreover, the DPSWFs can be computed by using two schemes. The first scheme is given by (5),that is an expansion with respect to the eigenvectors of the Toeplitz matrix ρ N,W , given by (4). Notethat from [17] and [22], the matrix ρ N,W is the matrix representation of I N B W I ∗ n , a composition ofindex- and band-limiting operators, I N : (cid:96) ( C ) → C N , B W : (cid:96) ( C ) → (cid:96) ( C ) , given for h = ( h n ) n ∈ Z , by B W ( h )( m ) = (cid:88) n ∈ Z sin(2 πW ( m − n )) π ( m − n ) h n , I N ( h )( m ) = h m , m ∈ { , . . . , N − } . This is a consequence of the connection between the DPSWF’s and their associated DPSS’s. Weshould mention that the DPSS’s are solutions of the following energy maximization dual problem.4et B W be the Paley-Wiener space given by B W = { h = ( h n ) n ∈ Z ∈ (cid:96) ( C ) , Supp ( (cid:98) h ) ⊂ [ − W, W ] } . Here, (cid:98) h ( x ) = (cid:88) n ∈ Z h n e − iπ ( N − − n ) x , x ∈ ( − / , / . Then, find h = arg max h ∈B W N − (cid:88) n =0 | h n | / (cid:16) + ∞ (cid:88) n = −∞ | h n | (cid:17) . Consequently, the DPSS’s are solutions of the system of equations (cid:88) m ∈ Z sin(2 πW ( m − n )) π ( m − n ) v ( k ) m = (cid:101) λ k,N ( W ) v ( k ) n , ∀ n ∈ Z . (18)For more details, see [17].The second scheme for the construction of the DPSWF’s is based on the computation of theeigenvectors of a Sturm-Liouville differential operator M W,N , commuting with the integral operator (cid:101) Q W,N , see for example [17]. This differential operator is given by M W,N ( g )( x ) = 14 π ddx (cid:20) (cos(2 πx ) − A ) ddx ( g )( x ) (cid:21) + 14 ( N −
1) cos(2 πx )( g )( x ) , A = cos(2 πW ) . (19)Hence, the DPSWFs are also given in terms with the eigenvectors of M W,N . It is easy to check that M W,N ( e iπ ( N − − n ) x )( x ) = 12 n ( N − n ) e iπ ( N − n +1) x + (cid:34) A (cid:18) N − − n (cid:19) (cid:35) e iπ ( N − n − x + 12 ( n + 1)( N − n − e iπ ( N − n − x (20)Consequently, the expansion coefficients in the basis { e iπ ( N − − k ) x , ≤ k ≤ N − } of the n -thDPSWFs ( U Nk,W ) are given by the components of the n -th eigenvector of the N × N tri-diadiagonalmatrix σ ( N, W ) , with coefficients given by σ ( N, W ) ij = i ( N − i ) , j = i − πW ) (cid:0) N − − i (cid:1) j = i ; ( i + 1)( N − i − , j = i + 1;0 , | j − i | > , i, j = 0 , . . . , N − . It is interesting to note that by considering the finite rank and positive-definite integral operator (cid:101) Q W,N as an operator acting on the Hilbert space L ( − W, W ) and by using the Min-Max theoremfor this operator, one gets the following lemma that provides us with a partial result related to thedecay rate of the (cid:101) Q W,N . We should mention that the proof of this lemma mimics the technique usedin [6] for proving a similar result concerning a decay rate of the λ n ( c ) , the eigenvalues of the operator Q c , given by (7) and associated with the classical PSWFs. Lemma 1.
For any real number < W < eπ and any integer N ≥ , we have (cid:101) λ n,N ( W ) ≤ C W √ N − (cid:16) neπW ( N − (cid:17) (cid:18) eπW ( N − n (cid:19) n − , eπW ( N − < n ≤ N − , (21) where C W = √ W (cid:16) eπW (cid:17) . roof: We first recall the Courant-Fischer-Weyl Min-Max variational principle concerning the pos-itive eigenvalues of a self-adjoint compact operator A acting on a Hilbert space H , with eigenvaluesarranged in the decreasing order λ ≥ λ ≥ · · · ≥ λ n ≥ · · · . In this case, we have λ n = min f ∈ S n max f ∈ S ⊥ n , (cid:107) f (cid:107) H =1 < Af, f > H , where S n is a subspace of H of dimension n. In our case, we have A = (cid:101) Q W,N , H = L ( − W, W ) . Weconsider the special case of S n = Span (cid:110) (cid:101) P ( x ) , (cid:101) P ( x ) , . . . , (cid:101) P n − ( x ) (cid:111) and f ( x ) = (cid:88) k ≥ n a k (cid:101) P k ( x ) ∈ S ⊥ n , (cid:107) f (cid:107) L − W,W ]) = (cid:88) k ≥ n | a k | = 1 . Here, (cid:101) P k ( x ) = (cid:114) k + 12 W P k (cid:16) xW (cid:17) , where P k is the usual Legendre polynomial of degree k and sat-isfying P k (1) = 1 . Note that the (cid:101) P k form an orthonormal family of L − W,W ) . The normalizationconstant follows from the fact that (cid:107) P k ( · ) (cid:107) L − W,W ) = (cid:32)(cid:90) W − W | P k (cid:16) yW (cid:17) | dy (cid:33) = (cid:18) W (cid:90) − | P k ( y ) | dy (cid:19) = (cid:114) W k + 1 = h k,W . (22)On the other hand, we have (cid:101) Q W,N (cid:16) (cid:101) P k (cid:17) ( x ) = (cid:90) W − W sin( N π ( x − y ))sin( π ( x − y )) (cid:101) P k ( y ) dy = (cid:90) W − W N − (cid:88) j =0 e iπ ( N − − j )( x − y ) h k,W P k (cid:16) yW (cid:17) dy = W N − (cid:88) j =0 e iπ ( N − − j ) x (cid:90) − e − iW π ( N − − j ) y h k,W P k ( y ) dy (23)Moreover, it is known that, see for example [15] (cid:90) − e ixy P k ( y ) dy = i k (cid:114) πx J k + ( x ) , x ∈ R . (24)where J α is the Bessel function of the first type and order α > . Further, the Bessel function J α has the following fast decay with respect to the parameter α , | J α ( z ) | ≤ (cid:12)(cid:12) z (cid:12)(cid:12) α Γ( α + 1) (25)Here, Γ( · ) is the Gamma function, that satisfies the following bounds, see [2] that √ e (cid:18) x + e (cid:19) x + ≤ Γ( x + 1) ≤ √ π (cid:18) x + e (cid:19) x + , x > − . (26)From the previous inequality and (22), we deduce that (cid:12)(cid:12)(cid:12) (cid:90) − e − iW π ( N − − j ) y h k,W P k ( y ) dy (cid:12)(cid:12)(cid:12) ≤ (cid:115) k + 1 W | N − − j | k + 1) (cid:18) eW π | N − − j | k + 1) (cid:19) k + (27)6hen, by using (23), (27) and the Minkowski’s inequality, one gets for k ≥ eπW ( N − / , (cid:13)(cid:13)(cid:13) (cid:101) Q W,N (cid:16) (cid:101) P k (cid:17) ( x ) (cid:13)(cid:13)(cid:13) L ( − W,W ) ≤ N − (cid:88) j =0 (cid:107) W e iπ ( N − − j ) x (cid:107) L ( − W,W ) (cid:115) k + 1 W | N − − j | k + 1) (cid:18) eW π | N − − j | k + 1) (cid:19) k + ≤ √ W N − (cid:88) j =0 (cid:115) eπW k + 1) (cid:18) eW π | N − − j | k + 1) (cid:19) k ≤ C W √ N − (cid:18) eW π ( N − k + 1) (cid:19) k + , C W = √ W (cid:16) eπW (cid:17) . (28)The last inequality follows from the fact that for k ≥ eπW ( N − / , N − (cid:88) j =0 | N − − j | k ≤ N − k + ( N − k +1 k + 1 ≤ ( N − k (cid:16) eπW (cid:17) . Hence, for the previous f ∈ S ⊥ n , and by using H¨older’s inequality, and taking into account that (cid:107) f (cid:107) L ( I,ω W ) = 1 , so that | a k | ≤ , for k ≥ n, one gets | < (cid:101) Q W,N f, f > L ([ − W,W ]) | ≤ (cid:88) k ≥ n | a k |(cid:107) (cid:101) Q W,N (cid:101) P k ( · ) (cid:107) L ([ − W,W ]) ≤ C W √ N − (cid:88) k ≥ n | a k | (cid:18) eW π ( N − k + 1) (cid:19) k + ≤ C W √ N − (cid:88) k ≥ n (cid:18) W eπ ( N − k + 1) (cid:19) k + . (29)The decay of the sequence appearing in the previous sum, allows us to compare this later with itsintegral counterpart, that is (cid:88) k ≥ n (cid:18) W eπ ( N − k + 1) (cid:19) k + ≤ (cid:90) + ∞ n − e − ( x + ) log ( x +1) eWπ ( N − ) dx ≤ (cid:90) + ∞ n − e − ( x + ) log ( neWπ ( N − ) dx (30)Hence, by using (29) and (30), one concludes thatmax f ∈ S ⊥ n , (cid:107) f (cid:107) L I,ωW )=1 < (cid:101) Q W,N f, f > L ([ − W,W ]) ≤ C W √ N − neW π ( N − ) e − ( n − ) log( neWπ ( N − ) . (31)To conclude for the proof of the lemma, it suffices to use the previous Courant-Fischer-Weyl Min-Maxvariational principle. (cid:3) In the first part of this section, we estimate the Hilbert-Schmidt norms of the two operators (2)and (7). As consequences, we give a comparison in the (cid:96) -norm of the spectrum associated with theDPSWFs with parameters N, W and the spectrum associated with the classical PSWFs ( ψ n,c ) n wih c = N πW.
Also, we give a fairly precise estimate of the number of the eigenvalues (cid:101) λ k,N ( W ) lyingin the interval [ ε, − ε ] , where ε ∈ (0 , / . In the second part, we use the energy maximizationscharacterizations of the DPSWFs and the PSWFs, and get an interesting fairly precise upper boundof the eigenvalues (cid:101) λ n,N ( W ) in terms of the eigenvalues λ n ( c ) , for 0 ≤ n ≤ N − . As a consequenceand by using the well established decay rates and behaviour of the λ n ( c ) , we deduce similar results7or the (cid:101) λ n,N ( W ) , with c = N πW.
The following proposition provides us with an (cid:96) -estimate of thespectrum of (cid:101) Q W,N by the spectrum of Q c . Note that since (cid:101) Q W,N is of finite rank, which is not thecase for the operator Q c , then this (cid:96) -estimate is done under the rule that (cid:101) λ k,N ( W ) = 0 , whenever k ≥ N. Proposition 1.
Under the previous notation, for W ∈ (0 , ) and an integer N ≥ , we have for c = πN W (cid:107) λ ( (cid:101) Q W,N ) − λ ( Q c ) (cid:107) (cid:96) = (cid:32) ∞ (cid:88) k =0 (cid:12)(cid:12)(cid:12)(cid:101) λ k,N ( W ) − λ k ( c ) (cid:12)(cid:12)(cid:12) (cid:33) ≤ W (cid:18) π W π ) (cid:19) (32) Proof:
Since the operator (cid:101) Q W,N acts on L ( − W, W ). Then we consider the operator Q W,c associatedwith the classical PSWFs that are mostly concentrated on [ − W, W ] and have bandwidth [ − c, c ].These last family of PSWFs are solutions of the eigenvalues problem Q W,c ( ψ ) = (cid:90) W − W sin( c ( x − y )) π ( x − y ) ψ k,W ( y ) dy = λ k,W ( c ) ψ k,W ( x ) . (33)It is well know that λ k,W ( c ) = λ k, ( cW ) = λ k ( cW ) , ∀ W > . It is common to write λ k, ( c ) = λ k ( c )and Q ,c = Q c , where this later is given by (7). For W ∈ (0 , ), we let c N = πN and c = πN W. Then, we have (cid:107) (cid:101) Q W,N − Q W,c N (cid:107) HS = (cid:90) W − W (cid:90) W − W (cid:18) sin( c N ( x − y ))sin( π ( x − y )) − sin( c N ( x − y ))( π ( x − y )) (cid:19) dxdy = W (cid:90) − (cid:90) − (sin( c ( t − u ))) (cid:20) W π ( t − u )) − W π ( t − u )) (cid:21) dudt (34)But for X = W π ( t − u ) ∈ [ − πW, πW ] , we have (cid:12)(cid:12)(cid:12)(cid:12) X − sin XX sin X (cid:12)(cid:12)(cid:12)(cid:12) ≤ | X | (cid:12)(cid:12)(cid:12)(cid:12) X sin X (cid:12)(cid:12)(cid:12)(cid:12) ≤ W π
W π sin(2
W π ) . (35)The last inequality is due to the fact that x (cid:55)→ x sin x is increasing on [ − πW, πW ] . Consequently,by using (34) and (35), one gets (cid:107) (cid:101) Q W,N − Q W,c N (cid:107) HS ≤ W (cid:18) W π πW ) (cid:19) . (36)Finally, by using (33) and the previous equality together with Wielandt-Hoffman inequality, onegets (cid:107) λ ( (cid:101) Q W,N ) − λ ( Q c ) (cid:107) (cid:96) = (cid:32) ∞ (cid:88) k =0 (cid:12)(cid:12)(cid:12)(cid:101) λ k,N ( W ) − λ k ( c ) (cid:12)(cid:12)(cid:12) (cid:33) ≤ W (cid:18) π W π ) (cid:19) . (cid:3) (37)Next by comparing the Hilbert-Schmidt norms of the operators (cid:101) Q W,N and Q c , together with aprecise estimate of T race ( Q c ) − (cid:107) Q c (cid:107) HS , we get the following theorem, showing that the eigenvalues (cid:101) λ k,W cluster around 1 and 0 . Theorem 1.
For any ε ∈ (0 , / and any W ∈ (0 , ) , let N ( W, ε ) = { k ; ε < (cid:101) λ k,N ( W ) < − ε } , then we have N ( W, ε ) ≤ π log(2 N W ) + 0 . − W + W c sin (2 c ) ε (1 − ε ) , c = πN W. (38)8 roof: Since (cid:107) (cid:101) Q W,N (cid:107) HS = (cid:90) W − W (cid:90) W − W (cid:18) sin( N π ( x − y ))sin( π ( x − y )) (cid:19) dxdy , then using the new variables t = xW , u = yW , we get (cid:107) (cid:101) Q W,N (cid:107) HS = W (cid:90) − (cid:90) − (cid:18) sin( πN W ( u − x ))sin( πW ( u − x )) (cid:19) dudx. That is for c = πN W, we have (cid:107) (cid:101) Q W,N (cid:107) HS − (cid:107)Q c (cid:107) HS = (cid:90) − (cid:18)(cid:90) − x − − x (sin( ct )) (cid:18) W sin ( πW t ) − t π (cid:19) dt (cid:19) dx = (cid:90) − (cid:18)(cid:90) − x − − x (sin( ct )) h W ( t ) dt (cid:19) dx (39)with h W ( t ) = W (cid:16) ( y ) − y (cid:17) = W g ( y ) and y ∈ ] − π, π [. We check that h W ( t ) ≥ W . Infact, g is even and increasing function on [0 , π ] . Note that straightforward computation gives us g (cid:48) ( y ) = y )) (cid:20) − cos( y ) + (cid:16) sin( y ) y (cid:17) (cid:21) . It is clear that g (cid:48) ( y ) ≥ y ∈ [ π , π ] , and (cid:18) sin( y ) y (cid:19) − cos( y ) ≥ (cid:16) − y (cid:17) − (cid:16) − y y (cid:17) ≥ y (cid:16) − y (cid:17) ≥ , y ∈ (cid:104) , π (cid:105) . Consequently, we have g ( y ) ≥ inf y ∈ [0 ,π ] g ( y ) = lim y → g ( y ) = 13 . (40)By combining (39) and (40), one gets (cid:107) (cid:101) Q W,N (cid:107) HS − (cid:107)Q c (cid:107) HS ≥ W (cid:90) − (cid:18)(cid:90) − x − − x − cos(2 ct )2 dt (cid:19) dx = 2 W − W c (cid:90) − [sin(2 ct )] − x − − x dx ≥ W − W c (sin(2 c )) On the other hand, from the proof of Lemma 2 of [3], it can be easily checked that (cid:107)Q c (cid:107) HS ≥ cπ − π log (cid:16) cπ (cid:17) − .
45 (41)By combining the previous two inequalities, one gets (cid:107) (cid:101) Q W,N (cid:107) HS ≥ cπ − π log (cid:0) cπ (cid:1) − .
45 + 2 W − W c (sin(2 c )) . Since
T race ( (cid:101) Q W,N ) = 2
N W = cπ , then by using the previous inequality, one gets T race ( (cid:101) Q W,N ) − (cid:107) (cid:101) Q W,N (cid:107) HS = N − (cid:88) k =0 (cid:101) λ k,N ( W )(1 − (cid:101) λ k,N ( W )) ≤ π log (cid:0) cπ (cid:1) + 0 . − W W c (sin(2 c )) . (42)That is for c = πN W, we have η ( N, W ) = N − (cid:88) k =0 (cid:101) λ k,N ( W )(1 − (cid:101) λ k,N ( W )) ≤ π log(2 N W ) + 0 . − W W c (sin(2 c )) . (43)9inally, since ∀ ε ∈ (0 , ) and x ∈ ( ε, − ε ) , we have x (1 − x ) ≥ ε (1 − ε ) , then ε (1 − ε ) N ( W, ε ) ≤ N − (cid:88) k =0 (cid:101) λ k,N ( W )(1 − (cid:101) λ k,N ( W )) ≤ η ( N, W ) . This conclude the proof of the theorem. (cid:3)
Remark 1.
We should mention that the upper bound given by (38) outperforms the bound given in[22] in the sense that N ( W, ε ) < π log( N −
1) + π N − N − ε (1 − ε ) , ∀ W ∈ (cid:16) , (cid:17) , N ≥ . Remark 2.
We should mention that our estimate of N ( W, ε ) , the number of eigenvalues in theinterval ( ε, − ε ) and given by (38) , is a non-asymptotic. It makes sense only if ε is not too small.Recently, in [11], the authors have given the following asymptotic estimate of N ( W, ε ) , which is validfor small values of ε, N ( W, ε ) = (cid:18) π log(8 N + 12) (cid:19) log (cid:18) ε (cid:19) . The following theorem is one of the main results of this work. It gives a fairly good bound of eacheigenvalue (cid:101) λ n,N ( W ) in terms the corresponding eigenvalue λ n ( c ) , with c = πN W and 0 ≤ n ≤ N − . This allows us to generalize at ounce the various existing upper bounds for the classical eigenvalues λ n ( c ) . Theorem 2.
Under the previous notation, for any integer N ≥ and real W ∈ (0 , / , we havefor c = N πW, (cid:101) λ n,N ( W ) ≤ A W λ n ( c ) , ≤ n ≤ N − , (44) where π ≤ A W = 2 π cos ( πW ) (cid:18) − W (cid:19) ≤ . (45) Proof:
We first use a classical technique for the construction of a subspace of the classical band-limited functions B Nπ = { f ∈ L ( − π, π ) , Supp t (cid:98) f ⊆ [ − π, (2 N − π ] } . This is done as follows. Let ϕ ( · ) ∈ L ( R ) , with Supp t (cid:98) ϕ ⊆ [ − π, π ] and let V N,ϕ = Span (cid:8) e iπkt ϕ ( t ) , ≤ k ≤ N − (cid:9) . That is if f ∈ V N,ϕ , then f ( t ) = N − (cid:88) k =0 (cid:98) P ( k ) e iπkt ϕ ( t ) . Here, N − (cid:88) k =0 (cid:98) P ( k ) e iπkt = P ( e iπt ) , where P ∈ R N − [ x ] is a polynomial of degree N − . Since Supp t ϕ ⊆ [ − π, π ] and since (cid:98) f ( ξ ) = N (cid:88) k =0 (cid:98) P ( k ) (cid:98) ϕ ( ξ − πk ) , then Supp t (cid:98) f ⊆ [ − π, (2 N − π ] , that is f ∈ B Nπ . By using Plancherel’s equality, one gets (cid:107) f (cid:107) L ( R ) = 12 π (cid:107) (cid:98) f (cid:107) L ( R ) = 12 π N − (cid:88) k =0 | (cid:98) P ( k ) | (cid:107) (cid:98) ϕ (cid:107) L ( R ) . Also, from Parseval’s equality, we have N − (cid:88) k =0 | (cid:98) P ( k ) | = (cid:107) P ( e iπt ) (cid:107) L ( − / , / .
10y combining the previous two equalities, one gets (cid:107) f (cid:107) L ( R ) = 12 π (cid:107) P ( e iπt ) (cid:107) L ( − / , / (cid:107) (cid:98) ϕ (cid:107) L ( R ) , deg P ≤ N − . On the other hand, for W ∈ (0 , / , we have (cid:107) f (cid:107) L ( − W,W ) = (cid:90) W − W | P ( e iπt ) | ϕ ( t ) dt ≥ min t ∈ [ − W,W ] | ϕ ( t ) | (cid:90) W − W | P ( e iπt ) | dt. Hence, for any f ∈ V N,ϕ , we have1min t ∈ [ − W,W ] | ϕ ( t ) | (cid:107) f (cid:107) L ( − W,W ) (cid:107) f (cid:107) L ( R ) ≥ π (cid:107) P ( e iπt ) (cid:107) L ( − W,W ) (cid:107) P ( e iπt ) (cid:107) L ( − / , / (cid:107) (cid:98) ϕ (cid:107) L ( R ) . (46)In particular, for (cid:98) ϕ ( ξ ) = [ − π,π ] ( ξ ) cos( ξ/ , we have (cid:107) (cid:98) ϕ (cid:107) L ( R ) = π. Moreover, we have ϕ ( x ) = 12 π (cid:90) π − π e ixξ cos( ξ/ dξ = π (cid:16) cos( πx ) − x (cid:17) , x ∈ ( − / , / if x = ± . So that min x ∈ [ − W,W ] | ϕ ( x ) | = 14 π (cid:18) cos( W π ) − W (cid:19) . Hence, for this choice of ϕ and by using (46), one concludes that for any polynomial P N ∈ R N − [ x ]of degree N − , we have (cid:107) P N ( e iπt ) (cid:107) L ( − W,W ) (cid:107) P N ( e iπt ) (cid:107) L ( − / , / ≤ (cid:18) − W cos( W π ) (cid:19) (cid:107) f N (cid:107) L ( − W,W ) (cid:107) f N (cid:107) L ( R ) , (47)where, f N ( t ) = P N ( e iπt ) ϕ ( t ) ∈ V N,ϕ . Next, let S N be the subspace of sequences xxx = ( x n ) n ∈ l ( C )with elements indexed on [[0 , N − , so that (cid:98) xxx ( t ) = N − (cid:88) k =0 x k e iπkt . Also, we denote by s n , v n , the( n + 1) − dimensional subspace of S N and V N,ϕ , respectively. Note that the eigenvalues of the Sinc-kernel operator, are invariant under dilation of the time-concentration interval and translation anddilation of the bandwidth concentration interval. That is for τ, c > , we have λ ( Q τ,c ) = λ ( Q ,τc ) = λ ( Q τc ) or equivalently λ n,τ ( c ) = λ n, ( τ c ) = λ n ( τ c ) . By using the previous properties as well as theMin-Max characterisation of this later, together with inequality (47), and the fact that V N,ϕ is asubspace of B ( N +1) π , one gets (cid:101) λ n,N ( W ) = max S n min xxx ∈ S n \{ } (cid:107) (cid:98) xxx (cid:107) L ( − W,W ) (cid:107) (cid:98) xxx (cid:107) L ( − , ≤ A W max v n min f ∈ v n \{ } (cid:107) f (cid:107) L ( − W,W ) (cid:107) f (cid:107) L ( R ) ≤ A W max U n min f ∈ U n \{ } (cid:107) f (cid:107) L ( − W,W ) (cid:107) f (cid:107) L ( R ) ≤ A W max W n min f ∈ W n \{ } (cid:107) f (cid:107) L ( − , (cid:107) f (cid:107) L ( R ) = A W λ n ( c ) . Here, A W = 2 π cos ( πW ) (cid:18) − W (cid:19) , U n is an ( n + 1) − dimensional subspace of B Nπ and W n is an( n + 1) − subspace of B c = { f ∈ L ( R ) , Supp t (cid:98) f ⊆ [ − c, c ] } , c = πN W. (cid:3) λ n ( c ) to theeigenvalues (cid:101) λ n,N ( W ) . In [6], it has been shown that for any c > n ≥ max (cid:16) , ec (cid:17) , we have λ n ( c ) ≤ e − (2 n +1) log( ec ( n +1)) . By using the previous theorem, together with the previousnon-asymptotic estimate of the λ n ( c ) , one concludes that for any N ≥ W ∈ (cid:0) , eπ N − N (cid:1) , we have (cid:101) λ k,N ( W ) ≤ e − (2 k +1) log( eπNW ( k +1)) , ≤ eπ N W ≤ k ≤ N − , (48)for any N ≥ W ∈ (cid:0) , eπ N − N (cid:1) . Moreover, it has been shown in [6] that for any cπ +log( c )+6 ≤ n ≤ c, there exists a uniform constant η > λ n ( c ) ≤ exp (cid:18) − η n − cπ log( c ) + 5 (cid:19) . This lastestimate combined with the previous theorem, give us the following similar estimate (cid:101) λ n,N ( W ) ≤ e − η n − NW log( πNW )+5 , N W + log( πN W ) + 6 ≤ n ≤ πN W. (49)It is interesting to note that besides providing an explicit exponential decay rate for the (cid:101) λ n,N ( W ) , the estimate (48) provides us with estimates for the unknown constants C ( W, ε ) , C ( W, ε ) appearingin the following asymptotic decay rate, given in [17] (cid:101) λ k,N ( W ) ≤ C ( W, ε ) e − C ( W,ε ) N , ∀ k ≥ (cid:100) N W (1 + ε ) (cid:101) , N ≥ N ( W, ε ) . (50)More precisely, by comparing (48) and (50), one concludes that for N ≥ , ε > eπ − and W ≤ eπ N − N , we have C ( W, ε ) ≤ , C ( W, ε ) ≥ W (1 + ε ) log (cid:18) ε ) + 2 eπ (cid:19) . (51) In this paragraph, we give two applications of the DPSWF’s. The first application is related to alower bound estimate for the constant appearing in the Tur`an-Nazarov concentration inequality, see[14]. The second applications deals with the quality of approximation by the DPSWFs of Bandlim-ited functions and functions from periodic Sobolev spaces.Let us first recall the following Tur`an-Nazarov type concentration inequality. Let T be the unitcircle and let µ be the Lebesgue measure on T , normalized so that µ ( T ) = 1 , then for every 0 ≤ q ≤ , every trigonometric polynomial P ( z ) = n +1 (cid:88) k =1 a k z α k , a k ∈ C , α k ∈ N , z ∈ T , and every measurable subset E ⊂ T , with µ ( E ) ≥ , we have (cid:107) P (cid:107) L q ( E ) ≥ e − A n µ ( T \ E ) (cid:107) P (cid:107) L q ( T ) . (52)Here, A is a constant independent of q, E and n. Since, the DPSWFs U Nn,W are given by U Nn,W ( x ) = (cid:15) n N − (cid:88) k =0 v nk ( e − iπx ) N − − k = P N ( e − iπx ) , x ∈ [ − / , / , q = 2 and E = ( − W, W ) where 1 / ≤ W < /
2, onegets (cid:101) λ n,N ( W ) = (cid:13)(cid:13) U Nn,W (cid:13)(cid:13) L ( − W,W ) (cid:13)(cid:13)(cid:13) U Nn,W (cid:13)(cid:13)(cid:13) L ( − / , / ≥ e − A (1 − W )( N − , ∀ ≤ n ≤ N − . (53)In particular, for n = N − , W = and by using the estimate (48), together with a straightforwardcomputation, one gets A ≥ − / (cid:18) eπ (cid:19) = 1 . . (54)Concerning the quality of approximation of bandlimited functions by the DPSWF’s, we havea partial result. In fact, we check that under some conditions on W and N, there exists N ≥ [2 N W ] − , such that the eigenspace spanned by the first N dilated DPSWFs √ W U
Nk,W ( W · )approximates the eigenspace spanned by the corresponding classical ψ k,c . For this purpose, we firstrecall the following result given by Theorem 3 of [23] and concerning the approximation of eigenspacesspanned by a set of eigenfunctions of positive self-adjoint Hilbert-Schmidt operator and its positiveself-adjoint perturbed version. More precisely, if A is such an operator with simple eigenvalues λ > λ > · · · and if there exists an integer D > λ D > δ D = ( λ D − λ D +1 ) andif A + B is such a perturbed operator satisfying the extra condition that (cid:107) B (cid:107) < δ D / , then (cid:107) π D ( A ) − π D ( A + B ) (cid:107) ≤ (cid:107) B (cid:107) δ D . (55)Here, π D ( A ) denotes the orthogonal projection over the space spanned by the first eigenfunctions ofthe operator A. In the sequel, we let (cid:101) L W,N denote the operator defined on L ( − ,
1) by (cid:101) L W,N ( f )( x ) = (cid:90) − sin( πN W ( x − y ))sin( πW ( x − y )) f ( y ) dy, x ∈ ( − , . (56)Then it is easy to check that the N dilated DPSWF’s √ W U
Nk,W ( W · ) are eigenfunctions of (cid:101) L W,N , with the same associated eigenvalues (cid:101) λ k,N ( W ) as the usual DPSWF’s. In the special case wherefor c = πN W, the operators A and A + B are given by Q c , and (cid:101) L W,N , respectively, we obtain thefollowing proposition that gives us an approximation of eigenspaces spanned by classical PSWFsand the corresponding DPSWFs. Proposition 2.
Let π K and (cid:101) π K be the two projection operators on the spaces spanned by the first K -eigenfunctions of the the operator Q c and (cid:101) L W,N , respectively. For any real b > log 3 π , there exists c b > such that for any ( W, N ) ∈ (cid:0) , (cid:1) × N , with c b ≤ πN W ≤ exp (cid:18) α b sin 2 πWW − − πb (cid:19) , α b = 332 bπ (cid:16) −
31 + e πb (cid:17) , (57) then there exists N ≥ [2 N W ] such that (cid:107) π N − (cid:101) π N (cid:107) ≤ W (cid:18) bπ πW ) (cid:19) log( πN W ) + 2 log 2 + πb − e πb . (58) Proof:
We first recall that in [18] and for a fixed b ≥ , c > , the author has given the followinglimit result for λ n ( c ) , lim c → + ∞ λ n c,b ( c ) = 11 + e πb , n c,b = (cid:20) cπ + 2 bπ log 2 + bπ log c (cid:21) (59)13ence, by applying the previous estimate for the two fixed values of b = 0 and b > log 3 π , oneconcludes that there exists C b > c ≥ c b , we have0 < n c,b − (cid:88) k = n c, λ k ( c ) − λ k +1 ( c ) = λ n c, ( c ) − λ n c,b ( c ) ≤ −
32 11 + e πb . (60)Note that from (59), we have n c,b − n c, = (cid:20) cπ + 2 bπ log 2 + bπ log c (cid:21) − (cid:20) cπ (cid:21) ≤ bπ log c + 2 bπ log 2 + 1 . Consequently, by using (60), one concludes there exists N ≥ n c, such that δ N = λ N ( c ) − λ N +1 ( c )2 ≥ bπ log c + bπ log 2 + 1 (cid:18) −
34 11 + e πb (cid:19) . Next, we consider the special cases of c = πN W, and the operators A and A + B are given by Q c , (cid:101) L W,N , respectively. By using (36) and (57), one can easily check that (cid:107) B (cid:107) = (cid:107) (cid:101) L W,N − Q c (cid:107) ≤ (cid:107) (cid:101) L W,N − Q c (cid:107) HS ≤ δ N . Hence by using (55) and (36), one gets the desired result (58). (cid:3)
It is well known see for example [16, 19], that if f ∈ B c , where B c is the space of bandlimitedfunctions, given by (6), then we have (cid:107) f − π N f (cid:107) L ( − , ≤ λ N ( c ) (cid:107) f (cid:107) L ( R ) . (61)Here, π N is the orthogonal projection over the first classical PSWFs ψ k,c ( · ) . Remark 3.
By combining (58) and the previous inequality, one gets the following partial resultconcerning the quality of approximation of bandlimited functions by the dilated DPSWF’s. For c = πN W, b > log 3 π and under condition (57) , there exists N ≥ [2 N W ] such that for f ∈ B c , wehave (cid:107) f − (cid:101) π N f (cid:107) L ( − , ≤ W (cid:18) bπ πW ) (cid:19) log( πN W ) + 2 log 2 + πb − e πb (cid:107) f (cid:107) L ( − , + λ N ( c ) (cid:107) f (cid:107) L ( R ) . (62) Here, (cid:101) π N is the orthogonal projection over the first N dilated DPSWFs √ W U
Nk,W ( W · ) . In a similarmanner, we may extend this approximation quality of the dilated DPSWF’s in the more general classof functions of almost time- and band-limited functions. For more details on this class of functions,the reader is refereed to [8, 10]. We leave the details of this extension to the reader.
Next, check that the DPSWFs are well adapted for the spectral approximation of functions fromthe periodic Sobolev spaces. Note that for a given real number s > , the periodic Sobolev space H sper ([ − / , / H sper ( − / , /
2) = (cid:40) f ∈ L ( − / , / , (cid:107) f (cid:107) H s = (cid:88) n ∈ Z (1 + n ) s | ˆ f n | < + ∞ (cid:41) , where (cid:98) f n = (cid:90) / − / f ( x ) e − iπnx dx. emma 2. let W ∈ (0 , ) , N ∈ N such that πN W ≥ . Let s > , and H sper ( − / , / , then thereexists a constant M > such that for any integer [2 N W ] + log( πN W ) + 6 ≤ K ≤ N − , we have (cid:107) f − (cid:101) π K ( f ) (cid:107) L ( − W,W ) ≤ N ) s/ (cid:107) f (cid:107) H s + (cid:113)(cid:101) λ K,N ( W ) (cid:107) f (cid:107) L ( − / , / . (63) Here, (cid:101) π K is the orthogonal projection over the space spanned by the first K DPSWFs, associatedwith the parameters
W, N.
Proof:
We first note that if f N ∈ L ( − / , /
2) is the function given by f N ( x ) = [( N − / (cid:88) k = − [( N − / (cid:98) f k e iπkx , then we have (cid:107) f − f N (cid:107) L ( − / , / = + ∞ (cid:88) | n |≥ [( N +1) / n ) s (1 + n ) s | ˆ f n | ≤ N ) s (cid:107) f (cid:107) H s . (64)On the other hand, by using the expressions of the DPSWFs, given by (5), as well as their doubleorthogonality property (3), on gets f N ( x ) = N − (cid:88) k =0 β k U Nk,W ( x ) , ∀ x ∈ [ − / , / , f N ( x ) = N − (cid:88) k =0 α k U Nk,W ( x ) (cid:113)(cid:101) λ k,N ( W ) , ∀ x ∈ [ − W, W ] . (65)Since the previous two expansions coincide on [ − W, W ] , then we have α k = β k (cid:113)(cid:101) λ k,N ( W ) (66)Moreover by using the previous identity, together with Parseval’s equality and the decay of the (cid:101) λ k,N ( W ) , one gets (cid:107) f N − (cid:101) π K ( f N ) (cid:107) L ( − W,W ) = N − (cid:88) k = K | α k | = N − (cid:88) k = K (cid:101) λ k,N ( W ) | β k | ≤ (cid:101) λ K,N ( W ) N − (cid:88) k = K | β k | ≤ (cid:101) λ K,N ( W ) |(cid:107) f (cid:107) L ( − / , / . (67)Moreover, since (cid:101) π K is an orthogonal projection, then we have (cid:107) (cid:101) π K (cid:107) ≤ . Hence, by using (64) and(67), one gets (cid:107) f − (cid:101) π K ( f ) (cid:107) L ( − W,W ) ≤ (cid:107) f − f N (cid:107) L ( − W,W ) + (cid:107) (cid:101) π K ( f − f N ) (cid:107) L ( − W,W ) + (cid:107) f N − (cid:101) π K ( f N ) (cid:107) L ( − W,W ) ≤ (cid:107) f − f N (cid:107) L ( − W,W ) + (cid:113)(cid:101) λ k,N ( W ) (cid:107) f (cid:107) L ( − / , / ≤ N ) s/ (cid:107) f (cid:107) H s + (cid:113)(cid:101) λ k,N ( W ) (cid:107) f (cid:107) L ( − / , / . (cid:3) (68) Remark 4.
It is easy to check that by considering the dilated DPSWF’s √ W U
Nk,W ( W · ) and byconsidering the periodic extension of f ∈ H s ( − , , s > , we get the following approximationresult of f by the first K dilated DPSWF’s, (cid:107) f − (cid:101) π K ( f ) (cid:107) L ( − , ≤ N ) s (cid:107) f (cid:107) H s + (cid:113)(cid:101) λ K,N ( W ) (cid:107) f (cid:107) L (cid:0) − / (2 W ) , / (2 W ) (cid:1) . (69)15 Numerical results.
In this section, we give three examples that illustrate the different results of this work.
Example 1:
In this first example, we give different numerical tests that illustrate the result ofProposition 1, implying in particular that each eigenvalue (cid:101) λ k,N ( W ) is well approximated by thecorresponding classical λ k ( c ) , c = πN W. Also, these tests illustrate the unexpected and importantinequality (44) of Theorem 2 that bounds each (cid:101) λ n,N ( W ) in terms of the corresponding λ n ( c ) , c = πN W and up to a small constant A W . This allows us also to check the exponential decay ratesfor the (cid:101) λ n,N ( W ) , given by (49) when n is close to the plunge region around [2 N W ] , and by (48)when n is sufficiently far from [2 N W ]. For these purposes, we have considered the value of N = 60and the four values of W = 0 . , . , . , . . Note that the (cid:101) λ n,N ( W ) are computed by computingthe eigenvalues of the Toeplitz matrix (4). The corresponding eigenvalues λ n ( c ) associated with theclassical PSWFs are computed with high precision by using the method given in [12]. In Table 1, wehave listed the (cid:96) -approximation error corresponding to both sequences of eigenvalues as predictedby proposition 1. In Figure 1(a), we have plotted the graphs of (cid:101) λ n,N ( W ) for the considered fourvalues of W. This figure illustrate Theorem 1 in the sense that the sequence (cid:101) λ n,N ( W ) clusters around1 and 0 and the number of the eigenvalues in the plunge region of the spectrum follows the boundgiven by Theorem 1. Finally, to illustrate the exponential decay rate of the (cid:101) λ n,N ( W ) as well as themain result of Theorem 2, we have plotted in Figure 1(b) the graphs of log (cid:0)(cid:101) λ n,N ( W ) (cid:1) versus thecorresponding log (cid:0) λ n ( c ) (cid:1) , c = πN W.W c = πN W (cid:107) λ ( Q c ) − λ ( (cid:101) Q W,N ) (cid:107) (cid:96) . .
85 4 . E − . .
70 1 . E − . .
55 3 . E − . .
40 8 . E − (cid:96) -error approximation of the sequence( (cid:101) λ n,N ( W )) n by the sequence ( λ n ( c )) n for N = 60 and different values of W. Example 2:
In this second example, we illustrate the quality of the spectral approximation ofbandlimited functions by the DPSWfs, as partially predicted by Proposition 2 and Remark 3. Forthis purpose, we have considered the α -bandlimited function f α defined by f α ( x ) = sin( αx ) αx with α = 56 and the special values of W = 0 . N = 60 , so that c N = πN W = 56 . . Then, wehave computed (cid:101) π N f α , the orthogonal projection of f α over the finite dimensional subspace spannedby the orthonormal set of L ( − ,
1) given by the dilated DPSWFs, ( √ W U
Nk,W ( W · )) ≤ k ≤ N − . Wefound that sup t ∈ [ − , | f α ( t ) − (cid:101) π N f α ( t ) | ≈ E − . That is (cid:101) π N f α provides us with a surprising high approximation of the f α . Example 3:
In this last example, we illustrate the quality of approximation of function from theperiodic Sobolev space H sper ( − ,
1) by the dilated DPSWFs, as predicted by Lemma 2 and Remark 3.16igure 1: (a) Graphs of the eigenvalues (cid:101) λ n,N ( W ) for the values of W = 0 . , . , . , . n. (b) Graphs of the associated log (cid:0)(cid:101) λ n,N ( W ) (cid:1) (in circles) versus thecorresponding log (cid:0) λ n ( c ) (cid:1) , c = πN W. (in boxes)For this purpose, we consider the Weierstrass function W s ( x ) = (cid:88) k ≥ cos(2 k x )2 ks , − ≤ x ≤ . (70)Note that W s ∈ H s − (cid:15)per ( − , , ∀ < (cid:15) < s. We consider the special value of s = 1 and the same setof dilated DPSWFs ( √ W U
Nk,W ( W · )) ≤ k ≤ N − , of the previous example with W = 0 . N = 60 . Then, we have computed the orthogonal projections (cid:101) π K W s , over the subspace spanned by the first K dilated DPSWFs. We found that (cid:107)W − (cid:101) π K W (cid:107) L ( − , ≈ . E − , K = N = 60an (cid:107)W − (cid:101) π K W (cid:107) L ( − , (cid:107) ≈ . E − , K = [2 N W ] = 36 . Note that this Weierstrass function has been already used in [5] to test the quality of approxima-tion of functions from the Sobolev spaces H s ( − ,
1) by the classical PSWFs ψ n,c . The previous twoapproximation errors and the numerical results given in [5], indicate that the DPSWFs outperformthe classical PSWFs for this kind of spectral approximation. In fact, with fewer expansion coeffi-cients, for this example, the DPSWFs provide better approximation of this Weierstrass function.This indicates that a DPSWF’s based scheme for the approximation of Sobolev spaces over compactintervals can be complementary to the proposed similar schemes based on the classical PSWFs, thathave been studied in [5, 19, 20].
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