# Polar wavelets and associated Littlewood-Paley theory

- 1996

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topEpperson Jay, and Frazier Michael. Polar wavelets and associated Littlewood-Paley theory. 1996. <http://eudml.org/doc/275829>.

@book{EppersonJay1996,

abstract = {Abstract
We develop an almost orthogonal wavelet-type expansion in ℝ² which is adapted to polar coordinates. We start by defining a product Fourier-Hankel transform f̂ and proving a sampling formula for f such that f̂ is compactly supported. For general f, the sampling formula and a partition of unity lead to an identity of the form $f = ∑_\{μ,k,m\}⟨f,φ_\{μ km\}⟩ψ_\{μkm\}$, in which each function $φ_\{μkm\}$ and $ψ_\{μkm\}$ is concentrated near a certain annular sector, has compactly supported product Fourier-Hankel transform, and is smooth away from the origin.
We introduce polar function spaces $A^\{α q\}_\{p\}$, analogous to the usual Littlewood-Paley spaces. We show that $A^\{02\}_\{p\} ≈ L^\{p\}$, 1 < p < ∞. We prove that $f ∈ A^\{α q\}_\{p\}$ if and only if a certain size condition on the coefficients $\{⟨f,φ_\{μkm\}⟩\}_\{μ,k,m\}$ holds. A certain class of almost diagonal operators is shown to be bounded on $A^\{αq\}_\{p\}$, which yields a product Fourier-Hankel transform multiplier theorem. Using this, we identify a polar potential operator $P^α$ which maps $A^\{βq\}_\{p\}$ isomorphically onto $A^\{α+β,q\}_\{p\}$.CONTENTS
1. Introduction and main results...............................................................5
2. Preliminaries.......................................................................................12
3. The sampling theorem and polar wavelet identity...............................25
4. Boundedness of almost diagonal matrices on $a^\{αq\}_\{p\}$...............27
5. Peetre’s maximal inequality.................................................................31
6. Norm characterizations.......................................................................35
7. FHT multiplier and potential operators................................................39
8. Equivalence of $L^p$ and $A^\{02\}_p$, 1 < p < ∞...............................42
9. Conclusion..........................................................................................49
References.............................................................................................50},

author = {Epperson Jay, Frazier Michael},

keywords = {polar wavelet identity; Peetre's maximal inequality; potential operators; orthogonal wavelet-type expansion; product Fourier-Hankel transform; sampling formula; polar function spaces; Littlewood-Paley spaces; multiplier},

language = {eng},

title = {Polar wavelets and associated Littlewood-Paley theory},

url = {http://eudml.org/doc/275829},

year = {1996},

}

TY - BOOK

AU - Epperson Jay

AU - Frazier Michael

TI - Polar wavelets and associated Littlewood-Paley theory

PY - 1996

AB - Abstract
We develop an almost orthogonal wavelet-type expansion in ℝ² which is adapted to polar coordinates. We start by defining a product Fourier-Hankel transform f̂ and proving a sampling formula for f such that f̂ is compactly supported. For general f, the sampling formula and a partition of unity lead to an identity of the form $f = ∑_{μ,k,m}⟨f,φ_{μ km}⟩ψ_{μkm}$, in which each function $φ_{μkm}$ and $ψ_{μkm}$ is concentrated near a certain annular sector, has compactly supported product Fourier-Hankel transform, and is smooth away from the origin.
We introduce polar function spaces $A^{α q}_{p}$, analogous to the usual Littlewood-Paley spaces. We show that $A^{02}_{p} ≈ L^{p}$, 1 < p < ∞. We prove that $f ∈ A^{α q}_{p}$ if and only if a certain size condition on the coefficients ${⟨f,φ_{μkm}⟩}_{μ,k,m}$ holds. A certain class of almost diagonal operators is shown to be bounded on $A^{αq}_{p}$, which yields a product Fourier-Hankel transform multiplier theorem. Using this, we identify a polar potential operator $P^α$ which maps $A^{βq}_{p}$ isomorphically onto $A^{α+β,q}_{p}$.CONTENTS
1. Introduction and main results...............................................................5
2. Preliminaries.......................................................................................12
3. The sampling theorem and polar wavelet identity...............................25
4. Boundedness of almost diagonal matrices on $a^{αq}_{p}$...............27
5. Peetre’s maximal inequality.................................................................31
6. Norm characterizations.......................................................................35
7. FHT multiplier and potential operators................................................39
8. Equivalence of $L^p$ and $A^{02}_p$, 1 < p < ∞...............................42
9. Conclusion..........................................................................................49
References.............................................................................................50

LA - eng

KW - polar wavelet identity; Peetre's maximal inequality; potential operators; orthogonal wavelet-type expansion; product Fourier-Hankel transform; sampling formula; polar function spaces; Littlewood-Paley spaces; multiplier

UR - http://eudml.org/doc/275829

ER -

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