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## Polar wavelets and associated Littlewood-Paley theory

Autorzy
Seria
Rozprawy Matematyczne tom/nr w serii: 348 wydano: 1996
Zawartość
Warianty tytułu
Abstrakty
EN
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 $A^{α+β,q}_{p}$.
EN
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
Słowa kluczowe
Tematy
Kategoryzacja MSC:
Miejsce publikacji
Warszawa
Seria
Rozprawy Matematyczne tom/nr w serii: 348
Liczba stron
51
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCXLVIII
Daty
wydano
1996
otrzymano
1995-03-02
Twórcy
autor
• Department of Mathematics, University of New Mexico, Albuquerque, New Mexico 87131, U.S.A.
autor
• Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, U.S.A.
Bibliografia
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Języki publikacji
 EN
Uwagi
1991 Mathematics Subject Classification: 42B25, 42C15.