Two–microlocal Besov spaces and wavelets

Moritoh, Shinya; Yamada, T.

doi:10.4171/rmi/389

Cited by 17 publications

(17 citation statements)

References 4 publications

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“…Unfortunately, the focus of the authors in those papers lies on the operators generated by such functions, but not on the functions itself. It would be desirable to clarify what kind of function spaces, maybe in the sense of microlocal analysis by Moritoh-Yamada [22] and Kempka [19] or even in the sense of varying smoothness [25], would be the right scale for these kernels. But that is not done within this work.…”

Section: Asymptotically Smooth Functionsmentioning

confidence: 99%

Error estimates for two-dimensional cross approximation

Schneider

2010

Journal of Approximation Theory

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In this paper we deal with the approximation of a given function f on [0, 1] 2 by special bilinear forms k i=1 g i ⊗ h i via the so-called cross approximation. In particular we are interested in estimating the error function f − k i=1 g i ⊗ h i of the corresponding algorithm in the maximum norm. There is a large amount of publications available that successfully deal with similar matrix algorithms in applied situations, for example in connection with H-matrices (see Boerm and Grasedyck (2003) [9] or Hackbusch (2007) [16] for many references). But as they do not give satisfactory error estimates, we concentrate on the theoretical issues of the problem in the language of functions. We connect it with related results from other areas of analysis in a historical survey and give a lot of references. Our main result is the connection of the error of our algorithm with the error of best approximation by arbitrary bilinear forms. This will be compared with the different approach in Bebendorf (2008) [6].

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Section: Asymptotically Smooth Functionsmentioning

confidence: 99%

Error estimates for two-dimensional cross approximation

Schneider

2010

Journal of Approximation Theory

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“…This notion can be extended to scales of spaces other than C s . For instance, the case of the spaces B s,p p is considered in [33,36].…”

Section: The Function D(x)mentioning

confidence: 99%

Wavelet techniques for pointwise regularity

Jaffard¹

2009

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…This notion can be extended to scales of spaces other than C s . For instance, the case of the Besov spaces B s,p p is considered in [24,27].…”

Section: One Immediately Checks That This Notion Is Weaker Than Höldementioning

confidence: 99%

“…This heuristic argument cannot be turned into a mathematical proof, and indeed, there exist counterexamples to (24) even when the spectrum of singularities is concave. When (24) holds, one says that f satisfies the p-multifractal formalism; the following result shows that it always yields an upper bound for the spectrum.…”

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confidence: 99%

Pointwise regularity associated with function spaces and multifractal analysis

Jaffard

2006

Approximation and Probability

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Abstract. The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) f has a given pointwise regularity exponent H. This notion has many variants depending on the global hypotheses made on f ; if f locally belongs to a Banach space E, then a family of pointwise regularity spaces C α E (x0) are constructed, leading to a notion of pointwise regularity with respect to E; the case E = L ∞ corresponds to the usual Hölder regularity, and E = L p corresponds to the T p α (x0) regularity of Calderón and Zygmund. We focus on the study of the spaces T p α (x0); in particular, we give their characterization in terms of a wavelet basis and show their invariance under standard pseudodifferential operators of order 0.

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Two–microlocal Besov spaces and wavelets

Cited by 17 publications

References 4 publications

Error estimates for two-dimensional cross approximation

Error estimates for two-dimensional cross approximation

Wavelet techniques for pointwise regularity

Pointwise regularity associated with function spaces and multifractal analysis

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