Continuous Wavelets and Frames on Stratified Lie Groups I

Geller, Daryl; Mayeli, Azita

doi:10.1007/s00041-006-6002-4

Cited by 44 publications

(59 citation statements)

References 27 publications

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“…(3) ( (3) was proved in [6], Lemma 7.6) In particular, B a /A a converges nearly quadratically to 1 as a → 1. For example, Daubechies calculated that if f (s) = se −s and a = 2 1/3 , then B a /A a = 1.0000 to four significant digits.…”

Section: Introductionmentioning

confidence: 95%

Besov spaces and frames on compact manifolds

Geller

Mayeli

2009

Indiana Univ. Math. J.

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Section: Introductionmentioning

confidence: 95%

Besov spaces and frames on compact manifolds

Geller

Mayeli

2009

Indiana Univ. Math. J.

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“…(5) and (6) were proved in [17], Lemma 7.6. In particular, B a /A a converges nearly quadratically to 1 as a → 1.…”

Section: Lemmas On Zonal Harmonicsmentioning

confidence: 99%

Spin Wavelets on the Sphere

Geller

Marinucci

2010

J Fourier Anal Appl

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121

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In recent years, a rapidly growing literature has focussed on the construction of wavelet systems to analyze functions defined on the sphere. Our purpose in this paper is to generalize these constructions to situations where sections of line bundles, rather than ordinary scalar-valued functions, are considered. In particular, we propose needlet-type spin wavelets as an extension of the needlet approach recently introduced by Narcowich et al. . We discuss localization properties in the real and harmonic domains, and investigate stochastic properties for the analysis of spin random fields. Our results are strongly motivated by cosmological applications, in particular in connection to the analysis of Cosmic Microwave Background polarization data.

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“…We began our program of looking at more general positive self-adjoint operators T , in our article [8]. There we took T to be the sublaplacian L on L 2 (G), where G is a stratified Lie group, and thereby obtained continuous wavelets and frames on such G.…”

Section: Introductionmentioning

confidence: 99%

Nearly tight frames and space-frequency analysis on compact manifolds

2008

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Let M be a smooth compact oriented Riemannian manifold of dimension n without boundary, and let be the Laplace-Beltrami operator on M. Say 0 = f ∈ S(R + ), and that f (0) = 0. For t > 0, let K t (x, y) denote the kernel of f (t 2 ). Suppose f satisfies Daubechies' criterion, and b > 0. For each j, write M as a disjoint union of measurable sets E j,k with diameter at most ba j , and measure comparable toform a frame for (I − P)L 2 (M), for b sufficiently small (here P is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for b sufficiently small). Moreover, based upon how well-localized a function F ∈ (I − P)L 2 is in space and in frequency, we can describe which terms in the summation F ∼ S F = j k F, φ j,k φ j,k are so small that they can be neglected. If n = 2 and M is the torus or the sphere, and f (s) = se −s (the "Mexican hat" situation), we obtain two explicit approximate formulas for the φ j,k , one to be used when t is large, and one to be used when t is small.

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Continuous Wavelets and Frames on Stratified Lie Groups I

Cited by 44 publications

References 27 publications

Besov spaces and frames on compact manifolds

Besov spaces and frames on compact manifolds

Spin Wavelets on the Sphere

Nearly tight frames and space-frequency analysis on compact manifolds

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