Paley-Wiener Approximations and Multiscale Approximations in Sobolev and Besov Spaces on Manifolds

Pesenson, Isaac Z.

doi:10.1007/s12220-008-9059-2

Cited by 20 publications

(31 citation statements)

References 35 publications

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“…The last item here follows from the general theory of frames [61]. We also note that for reconstruction of a Paley-Wiener vector from a set of samples one can use, besides dual frames, the variational (polyharmonic) splines in Hilbert spaces developed in [101]- [118].…”

Section: 5mentioning

confidence: 99%

Geometric Space–Frequency Analysis on Manifolds

Feichtinger

Führ

Pesenson

2016

J Fourier Anal Appl

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This paper gives a survey of methods for the construction of spacefrequency concentrated frames on Riemannian manifolds with bounded curvature, and the applications of these frames to the analysis of function spaces. In this general context, the notion of frequency is defined using the spectrum of a distinguished differential operator on the manifold, typically the Laplace-Beltrami operator. Our exposition starts with the case of the real line, which serves as motivation and blueprint for the material in the subsequent sections.After the discussion of the real line, our presentation starts out in the most abstract setting proving rather general sampling-type results for appropriately defined Paley-Wiener vectors in Hilbert spaces. These results allow a handy construction of Paley-Wiener frames in L 2 (M), for a Riemann manifold of bounded geometry, essentially by taking a partition of unity in frequency domain. The discretization of the associated integral kernels then gives rise to frames consisting of smooth functions in L 2 (M), with fast decay in space and frequency. These frames are used to introduce new norms in corresponding Besov spaces on M.For compact Riemannian manifolds the theory extends to Lp and Besov spaces. Moreover, for compact homogeneous manifolds, one obtains the socalled product property for eigenfunctions of certain operators and proves a cubature formulae with positive coefficients which allow to construct Parseval frames that characterize Besov spaces in terms of coefficient decay.Throughout the paper, the general theory is exemplified with the help of various concrete and relevant examples, such as the unit sphere and the Poincaré half plane.

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Section: 5mentioning

confidence: 99%

Geometric Space–Frequency Analysis on Manifolds

Feichtinger

Führ

Pesenson

2016

J Fourier Anal Appl

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“…Paley-Wiener spaces on manifolds play an important role in the construction and analysis of Besov spaces on various manifolds. See [11,33,7,23] for this direction of research.…”

Section: Introductionmentioning

confidence: 99%

What Is Variable Bandwidth?

Gröchenig

Klotz

2017

Comm Pure Appl Math

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Abstract. We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operatorf where p > 0 is a strictly positive function. Denote by c Λ (A p ) the orthogonal projection of A p corresponding to the spectrum of A p in Λ ⊂ R + , the range of this projection is the space of functions of variable bandwidth with spectral set in Λ.We will develop the basic theory of these function spaces. First, we derive (nonuniform) sampling theorems, second, we prove necessary density conditions in the style of Landau. Roughly, for a spectrum Λ = [0, Ω] the main results say that, in a neighborhood of x ∈ R, a function of variable bandwidth behaves like a bandlimited function with local bandwidth (Ω/p(x)) 1/2 .Although the formulation of the results is deceptively similar to the corresponding results for classical bandlimited functions, the methods of proof are much more involved. On the one hand, we use the oscillation method from sampling theory and frame theoretic methods, on the other hand, we need the precise spectral theory of Sturm-Liouville operators and the scattering theory of one-dimensional Schrödinger operators.

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“…Actually, for M+1 (ψ) 1 we have ψ − P M ψ 2 = 2 M+1 (ψ) ψ 2 ψ 2 , (4.53) so that, the reconstruction formula (4.5) for ψ M = P M ψ would give a good approximation of ψ , similarly to the approach followed in [27], Sect. 4.…”

Section: Proposition 412mentioning

confidence: 92%

“…[21,22]). However, a comprehensive study of the non-compact case is far more involved, although there is a quite well developed theory of sampling on Riemannian manifolds (see [11][12][13][24][25][26][27][28]) with reconstruction formulas for bandlimited functions on homogeneous spaces. Other results in this direction have been obtained for specific groups (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Sampling Theorem and Discrete Fourier Transform on the Hyperboloid

Calixto

Guerrero

Sánchez-Monreal

2010

J Fourier Anal Appl

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Using Coherent-State (CS) techniques, we prove a sampling theorem for holomorphic functions on the hyperboloid (or its stereographic projection onto the open unit disk D 1 ), seen as a homogeneous space of the pseudo-unitary group SU(1, 1). We provide a reconstruction formula for bandlimited functions, through a sinc-type kernel, and a discrete Fourier transform from N samples properly chosen. We also study the case of undersampling of band-unlimited functions and the conditions under which a partial reconstruction from N samples is still possible and the accuracy of the approximation, which tends to be exact in the limit N → ∞. Mathematics Subject Classification (2000)32A10 · 42B05 · 94A12 · 94A20 · 81R30 Communicated by T. Strohmer.M. Calixto ( ) · J.C. Sánchez-Monreal

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Paley-Wiener Approximations and Multiscale Approximations in Sobolev and Besov Spaces on Manifolds

Cited by 20 publications

References 35 publications

Geometric Space–Frequency Analysis on Manifolds

Geometric Space–Frequency Analysis on Manifolds

What Is Variable Bandwidth?

Sampling Theorem and Discrete Fourier Transform on the Hyperboloid

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