A note on spherical summation multipliers

Fefferman, Charles

doi:10.1007/bf02771772

Cited by 227 publications

(183 citation statements)

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“…This type of dilation corresponds to so-called parabolic scaling, which has a long history in the harmonic analysis literature and can be traced back to the "second dyadic decomposition" from the theory of oscillatory integrals [24,73] (see also the more recent work by Smith [72] on the decomposition of Fourier integral operators). It should be mentioned that, rather than A a , the more general matrices diag(a, a α ) with the parameter α ∈ (0, 1) controlling the "degree of anisotropy" could be used.…”

Section: Continuous Shearlet Systemsmentioning

confidence: 99%

Introduction to Shearlets

Kutyniok

Labate

2012

Applied and Numerical Harmonic Analysis

116

124

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Shearlets emerged in recent years among the most successful frameworks for the efficient representation of multidimensional data. Indeed, after it was recognized that traditional multiscale methods are not very efficient at capturing edges and other anisotropic features which frequently dominate multidimensional phenomena, several methods were introduced to overcome their limitations. The shearlet representation stands out since it offers a unique combinations of some highly desirable properties: it has a single or finite set of generating functions, it provides optimally sparse representations for a large class of multidimensional data, it is possible to use compactly supported analyzing functions, it has fast algorithmic implementations and it allows a unified treatment of the continuum and digital realms. In this chapter, we present a self-contained overview of the main results concerning the theory and applications of shearlets.

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Section: Continuous Shearlet Systemsmentioning

confidence: 99%

Introduction to Shearlets

Kutyniok

Labate

2012

Applied and Numerical Harmonic Analysis

116

124

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“…This dyadic-parabolic decomposition of phase space has been used in many papers to understand the L^R 71 ) behaviour of oscillatory integrals. We mention here the work of Fefferman [4] and Seeger-Sogge-Stein [8]; see also the presentation in chapter 9 of [12], where it is referred to as the second dyadic decomposition. We also mention the development by the author in [9] of a Hardy space based on this decomposition of phase space.…”

Section: The Frame Of Functionsmentioning

confidence: 99%

A parametrix construction for wave equations with $C^{1,1}$ coefficients

Smith¹

1998

Annales De L’institut Fourier

182

241

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“…In this case, the operator represents "standard" Bochner-Riesz means (referred to later as spherical means, as the set |ξ| = 1 is a sphere). In 1971, Fefferman [6] showed that for spherical means to be bounded on L p , it is necessary to have λ > λ * (p) = max{d| 1 2 − 1 p | − 1 2 , 0}. The Bochner-Riesz conjecture states that this is both necessary and sufficient.…”

Section: Introductionmentioning

confidence: 99%