TRIGONOMETRIC WAVELET INTERPOLATION IN BESOV SPACES. Woula Themistoclakis. 1. Introduction

Transcription

1 FACTA UNIVERSITATIS NIŠ) Ser. Math. Inform ), 49 7 TRIGONOMETRIC WAVELET INTERPOLATION IN BESOV SPACES Woula Themistoclakis Dedicated to Prof. M.R. Occorsio for his 65th birthday Abstract. In the resent aer we investigate a articular kind of trigonometric interolating wavelets, arising by an oortune discretization of the de la Vallée Poussin oerator. Wavelet comression is analyzed, and estimates of the interolating error are given in L 2π-norms and in Besov-norms.. Introduction Given a continuous 2π-eriodic function f, let us consider the Fourier sum of f: S m fx) := a m 2 + ) a k cos kx + b k sin kx, m N, where 2) a k := π 2π 2π k= ft)coskt dt, k =,,...,m, 3) b k := ft)sinkt dt, k =, 2,...,m. π It is well-known [2] that if we aly to the integrals 2) and 3), the following quadrature formula, of degree of exactness l: 2π T x) dx = 2π l ) 2kπ 4) T, l + l + Received May 6, Mathematics Subject Classification. 42A5, 42C5. 49

2 5 W. Themistoclakis choosing l := 2m, we obtain the trigonometric Lagrange olynomial: L mfx) := A m 2 + A k cos kx + B k sin kx, A k := B k := 2 2m + 2 2m + k= 2m j= 2m j= f f 2jπ 2m + ) cos 2jkπ 2m +, ) 2jπ sin 2jkπ 2m + 2m +, k =,,...,m, k =, 2,...,m, interolating f at the 2m + nodes θ m,k := 2kπ, k =,,...,2m. 2m + In this aer we aly a similar discretizing rocedure to the de la Vallée Poussin oerator: V m fx) := S mfx)+s m+ fx)+ + S 2m fx), m N. m Using the same quadrature formula 4), but with l :=, we obtain another interolating oerator, denoted by L m, which mas 2π-eriodic continuous functions into olynomials of degree 2m interolating on nodes, and which, like its continuous version V m, reserves the olynomials of order m. In the first art of this aer, we show that L m is a bounded ma in C2π and in some subsaces of L 2π. Several estimates of the interolation error are also given. In the second art of the aer, starting from an idea in [6], we consider a articular tye of trigonometric interolating scaling functions, defined as de la Vallée Poussin means of the Dirichlet kernels. Then we look the oerator L m from a different oint of view: as a rojector in the samled saces of a given eriodic multiresolution analysis [3]. This viewoint allows us to use all the tools of wavelet theory, i.e. decomosition, comression and reconstruction. Thanks to the multiresolution structure, given a 2π-eriodic, continuous function f, we can decomose the initial olynomial interolating f at resolution levels lower and lower. Then we can neglect that elements of the decomosition that are unimortant for the wanted aroximation of f and, finally, we can reconstruct the function at the initial resolution level. In this way we obtain a new comressed aroximation of f, that is no more interolating, but that is described by a number of decomosition coefficients less than the initial aroximation, aroximation degree being the same.

3 Trigonometric Wavelet Interolation in Besov Saces 5 Studying the error of the comressed aroximation, we indicate a criteria to choose the threshold for the comression. Finally we give some numerical examles, which confirm the following thesis: locally, in the regular arts of the function, the considered interolation rocess has a behaviour better than the olynomial of the best aroximation. In the resent aer only the trigonometric case is taken under consideration, but we advice that various results about the algebraic interolation can be deduced generalizing this trigonometric case in several directions. For examle, we can translate faithfully the trigonometric case banally, bringing about a change in variable x := cos t, t [,π]: in this way we still obtain interolating wavelets arising by a discretization of the de la Vallée Poussin algebraic oerator which is defined as mean of the Chebyshev sums see [7]). In [] interolating olynomial wavelets are constructed by means of a discretization of the Chebyshev sums, obtaining the classical Lagrange algebraic interolation on the zeros of the Chebyshev olynomials of the 2-th kind and on the extremes ±. In this case we can aly the decomositioncomression-reconstruction rocess, but we have no more uniformly convergent sequence of interolating olynomials. Nevertheless these are not the unique ossible aroaches to the algebraic case and we will return on this argument somewhere else. The outline of the resent aer is as follows. Section 2 is divided in two Subsections: in the first one, we give the notations and the definitions which are necessary in the sequel; in the second one, we discretize the de la Vallée Poussin oerator introducing the interolating oerator L m. In Section 3 we enunciate the main results about the boundedness of L m in the infinite norm and in Besov norms. In Section 4 we introduce the discrete version of the de la Vallée Poussin oerator by means of the wavelets theory. In Section 5 we analyze the effects of the decomosition-comression-reconstruction rocess alied to our interolating oerator, and finally, in Section 6 we give the roofs. 2. Preliminaries Some functional saces. Throughout this aer we denote by T n the set of all trigonometric olynomials of degree at most n and by C 2π the set of all continuous 2π-eriodic functions equied with the norm f := su fx). x [,2π)

4 52 W. Themistoclakis As usual the notation A B with A, B, means that there exist two ositive constants C,C 2 such that C B A C 2 B. Furthermore, throughout theaerwedenotebyc a ositive constant which may take different values in different formulas. For <+, L 2π is the set of 2π-eriodic functions f such that 2π ) / f := fx) dx < +, 2π while for = +, with L 2π we denote the set of 2π-eriodic functions everywhere defined and such that f < +. As usual, for each n N, + and f L 2π, E nf) denotes the error of the best trigonometric aroximation of f in the L 2π-norm, i.e., 5) E nf) := inf T T n f T. Furthermore, we denote by ω k f,t) the ordinary modulus of continuity of f, definedas 6) ω k f,t) := su k h f, ht with h fx) :=fx + h) fx), k h f := h k h f, h >, k N. The Jackson inequality: 7) E nf) Cω k f;/n), k < n, and the Salem-Stechkin inequality: ω k f; ) 8) C n n k n + i) k Ei f), hold for all n N, +, andf L 2π see [4]). Byusingω k f,t), we can define the following quasi-norm [ ] ωk f; t) q ) dt /q t r if q<+, t f,q,r := ω k f; t) su t> t r if q =+, i=

5 Trigonometric Wavelet Interolation in Besov Saces 53 for k>r, where, q, r are fixed real arameters such that + ; q + ; r. Adding to the above quasi-norm the L 2π-norm, we obtain the Besov norm: 9) f B r,q := f + f,q,r, and the corresonding Besov sace } Br,q := {f L 2π : f B r,q < +. By the inequalities 7) and 8), it can be roved that the Besov-norm 9) is equivalent to the norm defined by means of the error of the best trigonometric aroximation, as follows + ) /q [ + k) r /q Ek f) ] q, q<+, f E r,q := f + su + k) r Ek f), q =+. k Thus, ) f B r,q f E r,q, and the Besov sace Br,q admits a discrete equivalent version Er,q defined as } Er,q := {f L 2π : f E r,q < +. An interolating oerator. For each function f L 2π, let us consider the de la Vallée Poussin sum of f: V m fx) := m 2m n=m S n fx), m N, m 2, where S n f is the n + )-th artial sum of the Fourier series of f: S n fx) : = a n 2 + a k cos kx + b k sin kx, n N, k= a k : = π b k : = π 2π 2π ft)coskt dt, ft)sinkt dt, The following results are well-known [8]: k =,,...,n, k =, 2,...,n.

6 54 W. Themistoclakis i) For each T T m, V m T = T ; ii) V m := su V m f = 4 f = π 2 log 2 + c m, su m c m < +. Using the Dirichlet kernel D n x) :=+2 n cos kx, n N, k= we have the following integral reresentation V m fx) = 2π 2m ) ft) D n x t) dt. 2πm n=m A articular tye of trigonometric interolating olynomial can be deduced from ) by means of the following quadrature formula: 2) 2π T x) dx = 2π T 2kπ ), T T. In fact, alying 2) in ), we obtain the following trigonometric olynomial of degree 2m 3) L m fx) := [ m 2m n=m ] D n x t m,k ) ft m,k ), where t m,k := 2kπ, k =,,...,. Hence, taking into account that 2m 2, if x =2kπ, k Z, D n x) = sin x/2sinmx/2 n=m sin 2, otherwise, x/2 it is simle to recognize that the trigonometric interolating olynomial L m f interolates the function f at the nodes t m,k, k =,,...,. An exlicit exression of the olynomial L m f is the following 2m L m fx) =Ã 2 + λ k,m Ãk cos kx + B k sin kx), k=

7 Trigonometric Wavelet Interolation in Besov Saces 55 where, if k m, λ k,m := 2m k, m if k>m, for k =, 2,...,2m, and à k : = 2 ft m,j )coskt m,j, k =,,...,2m, j= B k : = 2 ft m,j )sinkt m,j, k =, 2,...,2m, j= t m,j : = 2πj, j =,,...,. We note that, by the well-made choice of the quadrature formula, the discrete oerator L m, like its continuous version V m, satisfies the following invariance roerty: T T m ) Lm T = T. 3. On the Boundedness of the Oerator L m First of all, the following theorem holds. Theorem. For each m N, <+, andf C 2π,itresults 4) ) / ) / A ft m,k ) L m f B ft m,k ), where t m,k =2kπ/), k =,,...,, and A = 3 3+4π, B =3+4π are suitable values of the constants A and B. The same result holds for =+, i.e., ) 3 3+4π max 2kπ k f L m f 3 + 4π) max 2kπ 5) k f ).

8 56 W. Themistoclakis This theorem can be found in [6], in a lightly different version. For convenience of the reader, in Section 6 we give a simle roof of it. Obviously, by Theorem we immediately deduce the following result: Corollary. 6) Let m N, +, andf C 2π.Then f L m f 4 + π)e mf) holds. We observe that for <+, 6) is not a homogeneous estimate by the resence of the infinite norm on the right hand side. A better estimate is given by the following statement. Theorem 2. For each such that + and for every function f C 2π satisfying ω k f; t) t +/ dt < +, we have f L m f C /m 7) ω k f; t) m / t +/ dt, where C is a constant indeendent of f and m. By Theorem 2, we obtain the following corollary. Corollary 2. Let, q + and r>/ arbitrarily fixed. Then for each function f C 2π Br,q, we have 8) f L m f C m r f B r,q, C being a constant indeendent of f and m. Finally, if we consider L m as a ma from the Besov sace B r,q into itself, we can state the following uniform boundedness result. Theorem 3. For each m N,, q +, s r, s>/, and f C 2π Bs,q, the following estimate 9) f L m f B r,q C m s r f B s,q

9 Trigonometric Wavelet Interolation in Besov Saces 57 holds, C being a constant indeendent of f and m. Consequently, for all, q + and r R + with r>/, 2) holds, where B r,q B r,q. su L m B m r,q B r,q < + is the sace B r,q C 2π, equied with the Besov norm 4. Interolating Scaling Functions and Wavelets Let us look at the interolating oerator L m from a different oint of view. For this end, from now on, let us consider m as a function of j N and more recisely, let us assume: m = m j := 2 j+, j N. For each j N, let for k =,,..., j ϕ j x) := 2m j 2 D n x), ϕ jk x) :=ϕ j x t jk ), t jk := 2kπ. j n=m j j It has been roved [6] that the functions ϕ j are scaling functions generating a eriodic Multiresolution Analysis shortly MRA) of L 2 2π, i.e., the subsaces of L 2 2π V j := san {ϕ jk : k =,,..., j }, j N, satisfy all roerties defining a eriodic MRA of L 2 2π see [3]): P) For each j N, V j V j+ ; P2) clos L 2 2π + j= V j = L 2 2π ; P3) j N, fx) V j = f P4) The set x 2π j ) V j ;

10 58 W. Themistoclakis {ϕ jk x) :=ϕ j x 2kπ ) } : k =,,... j j constitutes a Riesz basis of V j, i.e., it is a basis of V j and moreover, there exist two constants A, B > such that [ j ] /2 A a k 2 2) j j [ j 2 ] /2 a k ϕ jk B a k 2 j holds for each j N and each {a k } j C j. Furthermore, it has been shown that the following additional roerties hold: I) T mj V j T 2mj, foreachj N ; II) ϕ jk t jh )=δ hk,foreachj N and each h, k =,,..., j ; III) There exist two constants C,C 2 > such that for each j N, every {a k } j C j,andeach + ), it results 22) [ j ] / C a k j j For each j N, the function 23) ψ j x) :=2ϕ j+ x π j [ j a k ϕ jk C 2 j ) ϕ j x π j ) a k ] /. is the wavelet of level j see [6]). It is the unique function whose shifts ψ jk x) :=ψ j x t jk ), k =,,..., j, firstly, constitute a Riesz basis of the wavelet sace W j, defined by the conditions 24) V j+ = V j W j, V j W j, j N, and secondly, they satisfy the additional interolation roerty: ) 2h +)π ψ jk = δ hk. j

11 Trigonometric Wavelet Interolation in Besov Saces 59 The Riesz stability condition 22) holds for wavelet functions too, i.e. by 23), 22) we can deduce that there exist two constants, namely C and C 2, such that j N, {a k } j C j and + ), we have 25) C [ j ] / a k j j a k ψ jk C 2 [ j j ] / a k. Taking into account the interolation roerty of the scaling functions ϕ j roerty II), for each level j N, we can define the following interolating oerator: L j fx) := j ft jk )ϕ jk x); Lj ft jk )=ft jk ), k =,,..., j. This is nothing else than the interolating oerator L m in 3) calculated with m = m j =2 j+. But since obviously L j ϕ jk = ϕ jk, j N, k =,,..., j, we also have for each j N that f V j ) Lj f = f. Hence we can interret the discrete aroximation L mj = L j of the de la Vallée Poussin oerator V mj as a rojection on the samled sace V j too. This fact ermits us to aly the tyical strategies of wavelet theory, i.e. the decomosition, comression and reconstruction, as described in the next section. 5. Comression Fixed a function f L 2π and a resolution level J N, let us consider the aroximation f J of f at the level J, i.e. the interolating olynomial: 26) 27) f J x) := L J fx) = 2kπ a Jk := f J J a Jk ϕ Jk x), ), k =,,..., J,

12 6 W. Themistoclakis where, as usual, m J =2 J+. By virtue of 24), we have f J V J = V W W... W J. Hence f J can be decomosed as follows 28) f J x) = J a k ϕ k x)+ j= j b jk ψ jk x), where the decomosition coefficients {a k } and {b jk } in 28), can be calculated easily, starting from the initial coefficients {a Jk } in 27), by means of a decomosition algorithm that works in a very fast way, by using the FFT algorithm [6]. Once decomosed the initial aroximation f J of f, we can comress it, i.e. fixed an oortune threshold ε>, we can ut equal to zero those coefficients in 28) whose modulus is less than ε. In this way we obtain other decomosition coefficients: 29) {, if ak <ε, ã k := a k, otherwise, for k =,,...,m,and 3) bjk := {, if bjk <ε, b jk, otherwise, for k =,,...,m j, j =,...,J, by means of which we can reconstruct a new aroximation of f. Namely, 3) J x) := J ã k ϕ k x)+ f ε) j= j bjk ψ jk x) = J ã Jk ϕ Jk x). ε) We exlicitly note that also to reconstruct f J x) i.e. to calculate the coefficients {ã Jk } J, we can utilize a reconstruction algorithm that works recursively in a very simle and fast way. For more details we refer the reader to [6].

13 Trigonometric Wavelet Interolation in Besov Saces 6 Here we are interested to solve the following questions. How can we ε) choose the threshold ε in such a way as the comressed aroximation f J results comarable with the initial one f J? Furthermore, how many can we comress? That is, how many coefficients in 28) can we ut equal to zero? In Section 3 Corollary ) we have estimated the interolation error by means of the error of the best uniform aroximation: f f J CE m J f), +. Nevertheless locally, in the regular arts of f, the error fx) f J x) is smaller than E m J f). Consequently, taken ε E m J f), if the function is everywhere regular, we do not exect great results by the comression, but on the contrary, if the function f has some isolated oints of singularity, we exect to neglect a lot of coefficients in 28). At the moment we are not yet able to rove ointwise estimate for the error fx) f J x) confirming these facts, but several numerical tests give value to this idea. In the meantime, we state the following result: Theorem 4. For each J N,ε>and f C 2π,letf J and J be given by 26) and 3), resectively. Then, for all, +, itresults f ε) 32) f J ε) f J CJε, C being a constant indeendent of f, J, andε. Then, taking into account that f ε) f J f f J + f J f ε) J, by Theorem 4, easily we can deduce how to fix the value of the threshold ε in such a way as the aroximation degree does not change. In articular, choosing the threshold ε E m J f) 33), J by Theorem 4 and Corollary, we obtain the following estimate 34) f f ε) J CE m J f), +,

14 62 W. Themistoclakis where C is a constant indeendent of f and J. The following tables show some numerical results of the comression, for several resolution level J. Two cases are analyzed. In the first table, we have tested the function fx) = sin x for which Em J f) = O/m J )), while in the second table the fx) = sin x /2 is considered in this case Em J f) = O/ m J )). In the first case we have chosen ε =Jm J ), while in the second case we have taken ε =Jm /2 J ). In these tables we have denoted by N the number of non vanishing coefficients in 3), by N the number of vanishing coefficients in 3) and by P the ercentage of vanishing coefficients in 3). fx) = sin x. nodes N N P J = % J = % J = % J = % fx) = sin x /2. nodes N N P J = % J = % J = % J = % These results can be exlained with the following heuristic arguments. We have seen that the sequence f f j like the error of the best trigonometric aroximation E m j f). But if the function f is everywhere regular unless in some isolated oint, then E m j f) does not take into account the regular arts of f which, instead, should be sufficiently described by a lower resolution level. Then the more heterogeneous the function to aroximate is, the more successful the comression has. On the other side, for very regular function we can obtain a good error already with low resolution levels. Then it is obvious that for such functions we cannot exect great results about the comression in fact, for examle, if fx) =e sin x then we eliminate just two decomosition coefficients for all J =3, 4, 5, 6, 7, 8).

15 Trigonometric Wavelet Interolation in Besov Saces Proofs ProofofTheorem. The roof is based on the following trigonometric inequality see [8,. 228]): 35) ) ) / 2kπ T + 2πN ) T, which holds for all T T N, N N, and + for = + the left hand side denoting max T ) 2kπ k ). In fact, taking into account the interolation roerty of the oerator L m and alying the inequality 35) for T = L m f T 2m,wehave: ft m,k ) ) / = + + 4π 3 ) / L m ft m,k ) 2π2m ) ) L m f. ) L m f Hence, for A =3/3 + 4π), we obtain the first inequality in 4) and 5). In order to rove the second inequality in 4) and 5), let us start by considering the case <+. In this case there exists a unique function g L q 2π with / +/q =and g q =, such that L m f = 2π L m fx)gx) dx. 2π Then, by using the Hölder inequality and 35) with T = V m f T 2m,we have for <<+ the case = being analogous) L m f = 2π 2π L m fx)gx) dx = ft m,k )V m gt m,k ) ) /q V m gt m,k ) q ft m,k ) ) /

16 64 W. Themistoclakis + ) 2π2m ) V m g q + 4π ) V m q 3 3+4π 3 V m ft m,k ) ) / ft m,k ) ) / ft m,k ) ) / In the case =+, the same result can be roved by roceeding in the following way. For each x [, 2π), using the inequality 35) with =and T t) = 2m D n x t), we have n=m L m fx) 2m 2 ft m,k ) D n x t m,k ) n=m max k ft m,k ) 2m 2 D n x t m,k ) n=m ) ) 2π2m ) 2π 2m max ft m,k ) + D n x t) k 2πm dt n=m max ft m,k ) + 4π ) ) 2π 2m D n t) k 3 2πm dt, n=m and taking the suremum with resect to x [, 2π), we obtain: L m f 3+4π ) 2π 2m D 3 2πm n t) dt max ft m,k ). k n=m. But we recall that V m = 2π 2m D n t) 2πm dt n=m [ 2π 2m D n t) 2πm dt + n= 2π ] m D n t) dt n=

17 Trigonometric Wavelet Interolation in Besov Saces 65 ) 2π sin 2mt 2 2π = 2 sin mt ) 2 2πm sin t 2 dt + 2 sin t dt, 2 where we used the identity m D k t) = sin mt ) 2 2 sin t m N). 2 Then, since we can conclude 2π 2πm sin mt ) 2 2 sin t dt = m N), 2 V m 2+=3 m N). Hence, in both the cases <+ and =+, we obtain L m f 3+4π 3 V m [ max ft m,k ), k ft m,k ) ] /, <+, =+, 3 + 4π) [ max ft m,k ), k ft m,k ) ] /, <+, =+. ProofoftheCorollary. It is sufficient to consider the case =+. Let f C 2π and m N. Since obviously max k ft m,k ) f, by 5) we obtain 36) L m f 3 + 4π) f. Then the inequality 6) follows trivially by 36), recalling that L m is a quasi-rojector on T m, i.e., 37) T T m ) Lm T = T.

18 66 W. Themistoclakis In fact, for each T T m we have L m f f L m f T + T f = L m f T ) + T f 4 + 4π) f T, which gives 6) by taking the infimum with resect to T T m. ProofofTheorem2. Let be m N, + and f C 2π such that ω k f,t) t +/ dt < +. To rove 7) we use the following inequalities see [5,. 8 and. 69]) Ẽmf) Cτ k f; ) C /m ω k f,t) 38) m m / t +/ dt, where C is a constant indeendent of f and m, Ẽnf) is the error of the best one-sided trigonometric aroximation of f in the L 2π-norm, namely { Ẽnf) := inf T + T : T ± T n, T f T +}, and τ k f,t) denotes the τ-modulus of continuity, defined as τ k f,t) := ω k f,,t), with } ω k f,x,t) :=su { k h fs) : s, s + kh [, 2π] [x kt/2,x+ kt/2]. More recisely, by 37) and Theorem, for all T ± T m such that T f T +,wehave f L m f f T + L m f T T + T + L m f T ) ) / T + T + C f T )t m,k ) ) / T + T + C T + T )t m,k ) T + T + C L m T + T ) = C T + T.

19 Trigonometric Wavelet Interolation in Besov Saces 67 Thus, taking the infimum with resect to T ± T m such that T f T + and using 38), we get f L m f CẼ mf) Cτ k f; ) m C /m ω k f,t) m / t +/ dt. Proof of Corollary 2. Let be, q +, r>/, f C 2π B r,q, and m N, arbitrarily fixed. Let us examine the case q<+ the case q =+ being the same). If q =q )/q is the conjugate exonent of q, then by 7) and Hölder inequality, we obtain f L m f C m m ω k f; t) dt = C m t + m ω k f; t) [t q +r ]dt t r+ q C m ) m t [ q +r q ]q m dt [ ωk f; t) t r+ q ] q dt) q = C m m r f,q,r C m r f B r,q. Proof of Theorem 3. Let be m N,, q +, s r with s>/ and f C 2π Bs,q, arbitrarily fixed. First of all we observe that to demonstrate the theorem, it is enough to rove the following estimate 39) f L m E r,q C m s r f Es,q. In fact on the one hand, 39) imlies directly 9) by virtue of the normequivalence ), and, on the other hand, 2) trivially follows from 9) by taking s = r. Limit ourself to the case q +, the case q =+ being analogous. We note that T T m,wehave { f T if k<m, E k f T ) = E k f) if k m. Hence for T = L m f T 2m, we obtain f L m f E r,q = f L m f + + ) [ + k) r q Ek f L ] q q m f)

20 68 W. Themistoclakis f L m f ) [ + C + k) r q Ek f L ] q q m f) k<2m + C [ + k) r q Ek f L ] q m f) k 2m f L m f + C f L m f + C k 2m [ + k) r q E k f) C m r f L m f + k 2m k<2m q ] q q + k) rq ) q [ + k) r q E k f) q ] q. Then 39) follows immediately by Corollary 2 recall that s>/) and by the equivalence f B r,q f E r,q. In fact, we have f L m f E r,q [ C m r f L m f + + k) qr Ek f)q k 2m [ C m r f Bs,q m s + m qs r) [ f B s,q C m s r + m s r f Es,q ] ) q ] + k) qs Ek f)q k 2m C f E s,q m s r. ) q ] Proof of Theorem 4. Let be +, f C 2π, J N and ε> arbitrarily fixed. Let us write the initial aroximation of f at the level J 26), in the decomosition form 4) f J x) = J a k ϕ k x)+ j= j b jk ψ jk x). Then let us consider the corresonding comressed aroximation of f 4) f ε) J x) = J ã k ϕ k x)+ j= j bjk ψ jk x),

21 Trigonometric Wavelet Interolation in Besov Saces 69 with the coefficients {ã,k } and { b j,k }, resectively given by 29) and 3). If we set α,k := a,k ã,k ; k =,,...,, β j,k := b j,k b j,k ; k =,,..., j ; j =,,...,J, by 29) and 3), obviously we deduce that α,k <ε,,,...,, β j,k <ε,,,..., j ; j =,,...,J. Consequently, for all : +, it results 42) {α,k } l ε, {β j,k } l ε, j =,,...,J, where we have denoted by l {x k } l := N N max k the vectorial norm x k ) / <+, x k =+, where {x k } N C N. Thus, subtracting 4) from 4) and taking the -norm, by 22), 25) and 42), we get f J f ε) J J α,k ϕ k + j= j J C 2 {α,k } l + C 2 {β j,k } l εc 2 + C 2J) CJε. j= β j,k ψ jk Acknowledgement: The author is very grateful to Prof. G. Mastroianni for the helful suggestions and remarks. REFERENCES. T. Kilgore and J. Prestin, Polynomial wavelets on the interval. Constr. Arox ), 95.

22 7 W. Themistoclakis 2. G. Mastroianni Boundedness of Lagrange oerator in some functional saces. A survey. In: Aroximation theory and function series Budaest, 995), 7 39, Bolyai Soc. Math. Stud., 5, János Bolyai Math. Soc., Budaest, G. Plonka and M. Tasche, A unified aroach to eriodic wavelets. In: Wavelets: Theory, Algorithms, and Alications. Taormina, 993), 37 5, Wavelet Anal. Al., 5, Academic Press, San Diego, CA, P. P. Petrushev and V. A. Poov, Rational Aroximation of Real Functions. Encycloedia of Mathematics and its Alications. Cambridge University Press, V. A. Poov and B. Sendov, The Averaged Moduli of Smoothness. Alications in Numerical Methods and Aroximation. John Wiley and Sons, New York, J. Prestin and K. Selig, Interolatory and orthonormal trigonometric wavelets. In: Signal and Image Reresentation in Combined Saces, 2 255, Wavelet Anal. Al., 7, Academic Press, San Diego, CA, O. Kis and J. Szabados, On some de la Vallée Poussin tye discrete linear oerators, Acta Math. Hungar ), A. F. Timan, Theory of Aroximation of Functions of a Real Variable. Dover Publications, New York, 994. Istituto er Alicazioni del Calcolo M. Picone Sezione di Naoli C.N.R., Via P. Castellino 83 Naoli, Italy woula@iam.na.cnr.it