Polar wavelets and associated Littlewood-Paley theory

Epperson Jay; Frazier Michael

  • 1996

Abstract

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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 k m ⟩ ψ μ k m , in which each function φ μ k m and ψ μ k m 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 p α q , analogous to the usual Littlewood-Paley spaces. We show that A p 02 ≈ L p , 1 < p < ∞. We prove that f ∈ A p α q if and only if a certain size condition on the coefficients ⟨ f , φ μ k m ⟩ μ , k , m holds. A certain class of almost diagonal operators is shown to be bounded on A p α q , which yields a product Fourier-Hankel transform multiplier theorem. Using this, we identify a polar potential operator P α which maps A p β q isomorphically onto A p α + β , q .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 p α q ...............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 p 02 , 1 < p < ∞...............................42 9. Conclusion..........................................................................................49 References.............................................................................................50

How to cite

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Epperson 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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