Approximation of Integrable Functions by Wavelet Expansions

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1 Results Math 72 27, 23 2 c 26 The Authors. This article is published with open access at Springerlink.com /7/323-9 published online October 25, 26 DOI.7/s z Results in Mathematics Approximation of Integrable Functions by Wavelet Expansions W lodzimierz Lenski Bogdan Szal Abstract. Walter J Approx Theory 8:8 8, 995, Xiehua Approx Theory Appl 4:8 9, 998 Lal Kumar Lobachevskii J Math 342:63 72, 23 established results on pointwise uniform convergence of wavelet expansions. Working in this direction new more general theorems on degree of pointwise approximation by such expansions have been proved. Mathematics Subject Classification. 42C4, 4A25. Keywords. pointwise approximation, wavelet expansions.. Introduction In this paper we use the following notations. L p R<p< denotes the space of measurable, integrable with ph power functions f. The norm of f L p R is written by f Lp R. l 2 Z is the vector space of square-summable functions. A multiresolution approximation of L 2 R see[3 is a sequence V j j Z of closed subspaces of L 2 R such that the following hold: j Z V j V j, j Z V j is dense in L 2 R j Z V j, j Z fx V j f2x V j, j,k Z fx V j fx 2 j k V j, There exists an isomorphism I from V onto l 2 Z which commutes with the action of Z. We consider orthonormal bases of wavelets in L 2 R see[. These are functions of the form ψ j,k x 2 j 2 ψ 2 j x k,

2 24 W. Lenski B. Szal Results Math for some fixed function ψ are in turn based on a scaling function ϕ. In addition to the ladder of approximation spaces V V V L 2 R we have the orthogonal complement of each in the next higher one on the ladders i.e., V W V,V W V 2 The translates of the scaling function ϕ are an orthonormal basis of V while the translates of ψ are an orthogonal basis of W o see [. The same holds for their dilations in V j W j. Hence each f L 2 R has a representation where f x f n x f n x k j k j k f x f n xf n x, a n,k ϕ n,k x, ϕ n,k x 2 ϕ x k b j,k ψ j,k x, ψ j,k x 2 j 2 ψ 2 j x k b j,k ψ j,k x, with f n x n j k b j,k ψ j,k x, where a n,k, b j,k are expansion coefficients of f. f n is said to be the partial sums of the wavelet expansion. It is given f n x q n x, t f t dt by the reproducing kernels q n x, t ϕ n,k x ϕ n,k t p n x, t k p n x, t f t dt, n j k ψ j,k x ψ j,k t see [4. The pointwise modules of continuity of the function f at point x are given by ω x f,δ sup f t x f x, t δ δ w x f,δ f t x f x dt, δ δ for fixed x. The deviation f n x f x was estimated by Xiehua [, Theorem 3., p. 84 see also [5, Theorem 3. [ as follows:

3 Vol Approximation of Integrable Functions by Wavelet Expansions 25 Theorem A. Assume the scaling function ϕ satisfies the condition C ϕ x x α, α >. If f L 2 R is continuous at x, then we have f n x f x { O f L2R } α fx 2 α n k α ω x f,k2, where O depends only on C. k We also note the following well known result of Daubechies. Theorem B [2, Theorem Letϕ be a continuous function such that ϕ ϕ satisfy. Ifϕ n,k constitute an orthonormal basis on L 2 R, then the {ϕ n,k : n, k Z} also constitute an unconditional basis for all the spaces L p R <p<. Here, we will give more general precise estimates of the pointwise convergence of wavelet expansions. 2. Statement of the Results We determine the degree of approximation of functions by wavelet expansions. Theorem. Assume that a continuous function ϕ is such that ϕ ϕ satisfy. Iff L p R <p<, then } f n x f x K {2w x f,2 α α t α w x f,tdt 2 at every point x with a positive constant K dependent on C. Proof. From it is easy to say that K qx, t x α. 2 Following Walter [, Meyer [6 Novikov, Protasov, Skopina [7 we have f n x f x Next, using 2 q n x, t dt, for x R 3 q n x, t {f t f x} dt x2 x 2. x2

4 26 W. Lenski B. Szal Results Math x2 x 2 x2 x 2 [ K t α K x t α f t f x dt K t α [ f t x f x f x f x dt K [ d t t α f u x f x du dt dt t f u x f x du 2 αkα t α [ t [ t α 2 f u x f x du dt K α w xf,2 w x f,2 Kw x f,2. Observing that x2 2 x2 2 2 K α K α 2 t α K α [ t α t αk α t lim t t α f u x f x du { lim 2 p t α t 2 t αkα t α [ t α 2 dt K x t α f t f x dt K t α f t x f x dt f t x f x t α dt [ d dt f u x f x du dt f u x f x du 2 t α 2 [ t f u x f x du dt. [ t /p } f u x du p 2t α f x t } { lim 2 p t α p f Lp t R 2t α f x

5 Vol Approximation of Integrable Functions by Wavelet Expansions 27 we obtain x2 Kw xf,2 K α α Collecting our partial estimates the result follows. 2 t α w x f,tdt. Corollary. If f {g L p R :w x g, δ O w x δ} where w x is a function of modulus of continuity type, then by the monotonicity of wxδ δ α>2, we obtain f n x f x O w x 2. Theorem 2. Under the assumptions of Theorem, we have f n x f x 2Kw x f,2 Kα α t α w x f,tdt 2 [ K α 2 2 q f /q L αq p R 2 α f x, where p q. Proof. Similarly as above f n x f x q n x, t {f t f x} dt x2 x 2 Kw xf,2. Further x2 K α f t x f x f x f x 2 t α dt [ d t K α t α f u x f x du dt 2 2 dt [ t K α t α f u x f x du αk α 2 t α 2 [ t f u x f x du dt x2 x 2 x2 2

6 28 W. Lenski B. Szal Results Math K α [ 2 p f L p R 2 f x Kw x f,2 K α α t α w x f,tdt, 2 { K α t qα q dt R t qα q dt f x p dt R K α 2 αq f /q L p R Thus we obtain the desired estimate. f t x p p dt p 2 f x. α } 2 f x t α dt Corollary 2. Under the assumptions of Theorem if w x f,δ O δ γ, then by Theorem 2 O 2 γ when <γ<α, f n x f x O 2 α when γ>α, O n2 α when γ α. [9. The same order of approximation one can find in the paper of Skopina Remark. If x is a point of continuity of f L p R, then ω x f,δ o, whence w x f,δ o thus f n x f x o. We also show our estimates in the following summation forms. Theorem 3. Under the assumptions of Theorem f n x f x K α α ν 2 α 2 w x f, where K α K [ 2 α2 2α. 2αK ν ν ν α w x f, ν, ν Proof. Since δw x f,δ is nondecreasing function of δ, therefore for N N N t α w x f,tdt t α w x f,tdt 2 2 n u α u w x f, N2 ν2 du t α tw x f,tdt u ν ν2

7 Vol Approximation of Integrable Functions by Wavelet Expansions 29 ν ν ν ν ν 2 ν 2 ν 2 ν 2 2 ν ν w x u α u w x f, ν [ u α α f, du u ν uν N2 ν ν2 ν2 N2 ν w x ν ν w x f, N2 ν α ν w x f, ν ν 2 α 2 w x f, ν w x ν N2 ν α 2 α ν 2 α 2 w x f, ν 2 α 2 w x f, ν ν ν ν f, ν t α tw x f,tdt f, ν ν ν α w x N2 ν 2nα ν α w x ν ν 2 α ν 2 wx f, ν α ν α w x ν ν ν 2 α ν wx f, ν ν 2 α wx f, 2 wx 2 wx f, α 2 α 2nα w x f, Thus, by Theorem our result follows. Theorem 4. Under the assumptions of Theorem f n x f x K α where K α K [ 2 α2 2α. ν ν α w x f, ν [ t α α ν α 2 αn f, ν ν2 tν2 f, ν f, ν 2 f, n α 2 n. [ K α 2 2 q /q αq 2 α f x, f Lp R

8 2 W. Lenski B. Szal Results Math Proof. Using the monotonicity of δw x f,δ we obtain that ν 2 t α w x f,tdt ν ν ν ν2 ν2 ν w x t α tw x f,tdt f, ν ν2 t α dt ν2 [ t α ν2 ν w x f, ν ν α w xf, ν ν α ν α w x f, ν ν ν α ν ν α w x f, ν 2 αn w x α [ 2 n tν2 ν α 2 n α f, ν [ ν α ν α 2 α n ν ν α w x f, ν ν 2 α n ν α w x f, ν ν f, ν w x 2 α w x f, w x f,. 2 ν ν ν α Hence, by Theorem 2, analogously as above, our result follows. Remark 2. Since w x f,δ 2ω x f,δ, 4 we can immediately write all of the above estimations with ω x instead of w x. Remark 3. In the case p 2 our assumptions confine to the function ϕ only thus the mentioned result of Xiehua [, Theorem 3., p. 84 follows immediately from Theorem 4, by Remark 2. Remark 4. The convergence at Lebesgue s strong Lebesgue s points were investigated in the papers of Skopina [8 of Kelly, Kon, Raphael [4. We very thank the referee for helpful comments constructive suggestions.

9 Vol Approximation of Integrable Functions by Wavelet Expansions 2 Open Access. This article is distributed under the terms of the Creative Commons Attribution 4. International License by/4./, which permits unrestricted use, distribution, reproduction in any medium, provided you give appropriate credit to the original authors the source, provide a link to the Creative Commons license, indicate if changes were made. References [ Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. XLI, [2 Daubechies, I.: Ten lectures on wavelets. SIAM, Philadelphia 992 [3 DeVore, R.A.: The approximation of continuous functions by positive linear operators. Lecture Notes in Math., vol Springer, Berlin 972 [4 Kelly, S.E., Kon, M.A., Raphael, L.A.: Local convergence for wavelet expansions. J. Funct. Anal. 26, [5 Lal, S., Kumar, M.: Approximation of functions of space L 2 R by wavelet expansions. Lobachevskii J. Math. 342, [6 Meyer, Y.: Ondelettes et opérateurs, vol.. Herman, Paris 99 [7 Novikov, I.Ya., Protasov, V.Yu., Skopina, M.A.: Wavelet theory. AMS, Providence 2 [8 Skopina, M.A.: Local convergence of Fourier series with respect to periodized wavelets. J. Approx. Theory 94, [9 Skopina, M.A.: Wavelet approximation of periodic functions. J. Approx. Theory 4, [ Walter, G.G.: Pointwise convergence of wavelet expansions. J. Approx. Theory 8, [ Xiehua, S.: On the degree of approximation by wavelet expansions. Approx. Theory Appl. 4, W lodzimierz Lenski Bogdan Szal Faculty of Mathematics Computer Science Econometrics University of Zielona Góra ul. Szafrana 4a Zielona Gora Pol W.Lenski@wmie.uz.zgora.pl; B.Szal@wmie.uz.zgora.pl Received: May 7, 26. Accepted: October, 26.