A Bernstein-type inequality associated with wavelet decomposition

Abstract. We study tight wavelet frame systems in Lp (R d ) and prove that such systems (under mild hypotheses) give atomic decompositions of Lp(R d ) for 1 < p < ∞. We also characterize Lp(R d ) and Sobolev space norms by the analysis coefficients for the frame. We consider Jackson inequalities for best mterm approximation with the systems in Lp(R d ) and prove that such inequalities exist. Moreover, it is proved that the approximation rate given by the Jackson inequality can be realized by thresholding the frame coefficients. Finally, we show that in certain restricted cases, the approximation spaces, for best m-term approximation, associated with tight wavelet frames can be characterized in terms of (essentially) Besov spaces.

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“…By the result of Jia [22], for each 0 < α < s/d, the Bernstein inequality Thus, for j ∈ Z and k ∈ Z d , we have…”

Section: Moreover We Have the Jackson Inequalitymentioning

confidence: 99%

“…In general, this is an open (and likely very hard) problem but we conclude this paper by mentioning one important case where a Bernstein inequality can be proved. The proof relies heavily on the result by Jia [22].…”

Section: Moreover We Have the Jackson Inequalitymentioning

confidence: 99%

Tight wavelet frames in Lebesgue and Sobolev spaces

Borup

Gribonval

Nielsen

2004

Journal of Function Spaces

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“…In their work, the refinement mask was required to be nonnegative. In [15], Jia extended their results and, in particular, removed the restriction of non-negativity of the mask.…”

Section: Introductionmentioning

confidence: 99%

Approximation properties of multivariate wavelets

Jia

1998

Math. Comp.

Self Cite

131

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Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in W k−1 1 (R s ) provides approximation order k.

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“…The method is new and is simpler than those used earlier in the proof of that inequality (see [11]- [13]). It should be noted that the techniques of dyadic space enable us to obtain the result also for the couple (BMO,…”

Section: Definition 24 a Function S Is Said To Belong Tomentioning

confidence: 99%

On the computation of $K$-functionals

Irodova

2010

St. Petersburg Math. J.

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Abstract.A new approach to the calculation of the sharp order of a K-functional is suggested. This approach employs the techniques of dyadic spaces. §1. IntroductionThe theory of local polynomial approximation makes it possible to construct dyadic analogs of certain spaces. The dyadic Hardy classes and the dyadic BMO are primary examples of this sort, which have found interesting applications in analysis.In the author's paper (the definition of B λθ p (F ) will be given below). In [1] it was noted that there are at least two reasons for which the study of dyadic spaces is of interest. First, we deal with a new scale of spaces, and the functions belonging to them can be viewed as functions on a graph. Indeed, let us represent the dyadic family F as a tree. Its root vertex corresponds to the d-dimensional unit cube, and 2 d edges emanate from each vertex. Then the "weight" of each vertex is defined in terms of local approximation by polynomials on the cube corresponding to that vertex. This treatment of a space enables us to simplify the proofs, to make them more geometric, and to discard any restrictions on the integrability exponent p. It should be noted that the traditional approach to classical spaces, as developed by Nikol skiȋ and Besov (see, e.g., [2]), is based on integral representations and is not applicable for 0 < p < 1. So, dyadic spaces constitute a useful discretization of classical spaces and provide a useful language for the description of applied algorithms.Second, dyadic B-spaces are closely related to classical B-spaces. Compared to its classical counterpart, the dyadic space B . Thanks to this, many results for B-spaces can be deduced from similar results for dyadic spaces. Our objective in this paper is to show how the techniques of dyadic spaces can be used in classical analysis; this will be done by considering the calculation of a K-functional as an example.A different approach to the calculation of K-functionals was presented in [3]- [5]. The present paper can be regarded as a continuation of the author's paper [1]; it is organized as follows.

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A Bernstein-type inequality associated with wavelet decomposition

Cited by 24 publications

References 15 publications

Tight wavelet frames in Lebesgue and Sobolev spaces

Tight wavelet frames in Lebesgue and Sobolev spaces

Approximation properties of multivariate wavelets

On the computation of $K$-functionals

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