Besov spaces and frames on compact manifolds

Geller, Daryl; Mayeli, Azita

doi:10.1512/iumj.2009.58.3741

Cited by 35 publications

(77 citation statements)

References 22 publications

(82 reference statements)

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“…The theory was worked out in full generality in [18][19][20]. Specifically, one starts with the Calderón formula:…”

Section: Lemmas On Zonal Harmonicsmentioning

confidence: 99%

“…Spherical needlets were recently introduced by [38,39], and further developed, and extended to general smooth compact Riemannian manifolds in [18][19][20]. In a random fields environment, needlets were investigated by [5,6], with a view to applications to the statistical analysis of CMB data; applications in the physical literature include [26,34,42], see also [12,15,16,30] and [31,35].…”

Section: Introductionmentioning

confidence: 99%

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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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“…The theory was worked out in full generality in [18][19][20]. Specifically, one starts with the Calderón formula:…”

Section: Lemmas On Zonal Harmonicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Spin Wavelets on the Sphere

Geller

Marinucci

2010

J Fourier Anal Appl

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121

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“…In another article (already available [10]), we show that one can determine whether F is in a Besov space, solely from a knowledge of the size of its frame coefficients. In a future article, we hope to study the same question for Triebel-Lizorkin spaces.…”

Section: Introductionmentioning

confidence: 98%

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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“…Frames on non-Euclidean spaces were considered in [6,9,12,10,11,13,18,[21][22][23][24]. Applications of frames on manifolds to scattering theory, statistic and cosmology can be found in [10,11,17,18].…”

Section: Introductionmentioning

confidence: 99%