On Sampling Formulas on Symmetric Spaces

Ebata, Mitsuhiko; Eguchi, Masaaki; Koizumi, Shin; Kumahara, Keisaku

doi:10.1007/s00041-005-4014-0

Cited by 9 publications

(11 citation statements)

References 9 publications

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“…Some of the ideas and methods of this theory were recently extended to the cases of Riemannian manifolds, symmetric spaces, groups, and quantum graphs [7], [8], [10], [11], [14], [15], [25]- [32].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Sampling in Paley-Wiener spaces on combinatorial graphs

Pesenson

2008

Trans. Amer. Math. Soc.

232

244

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Abstract. A notion of Paley-Wiener spaces on combinatorial graphs is introduced. It is shown that functions from some of these spaces are uniquely determined by their values on some sets of vertices which are called the uniqueness sets. Such uniqueness sets are described in terms of Poincare-Wirtingertype inequalities. A reconstruction algorithm of Paley-Wiener functions from uniqueness sets which uses the idea of frames in Hilbert spaces is developed. Special consideration is given to the n-dimensional lattice, homogeneous trees, and eigenvalue and eigenfunction problems on finite graphs.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Sampling in Paley-Wiener spaces on combinatorial graphs

Pesenson

2008

Trans. Amer. Math. Soc.

232

244

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“…Together with (18), (19), and (21), this ensures the possibility of choosing q i so as to satisfy inequality (28).…”

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

“…Together with (18), (19), and (21), these inequalities show that q i can be chosen so as to satisfy (29) and (30). It should be mentioned that if summation in the numerators on the left in (29) and (30) is taken over some subsequence of indices (rather than over all s ≥ q i ), then these inequalities will remain valid.…”

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confidence: 90%

On divergence of sinc-approximations everywhere on $(0,\pi)$

Trynin

2011

St. Petersburg Math. J.

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Abstract. Some properties of sinc-approximations of continuous functions on a segment are studied.For the first time, the sinc-approximations arose in the work of Pleynet. Later, in connection with developments in signals coding theory, E. Borel and E. T. Whittaker introduced the notions of a cardinal function and a truncated cardinal function, the restriction of which to [0, π] looks like this:(1)By the present time, the problem of sinc-approximation has been studied rather deeply for functions analytic on the real line and decaying exponentially at infinity (see, e.g.,The most complete survey of the results obtained in this direction before 1993, together with numerous important applications of sinc-approximations, can be found in [6]. An interesting historical survey of investigations on this topic is contained in [7].Also, a series of publications had their origin in what is called the sampling theorem or sometimes, the Whittaker-Kotel nikov-Shannon discretization theorem [8,9,6,11]; in those studies, various representations of entire functions were obtained in the form of series in sincs with interpolation nodes satisfying some "uniform distribution" conditions (see, e.g., [12]-[14]). Starting with the well-known paper [15] by Kramer, the relationship between the sampling theorems and the Lagrange interpolation with nodes in the SturmLiouville problem spectrum has been studied; see, e.g., [16].The sinc-approximations are widely applied for constructing numerical methods of mathematical physics and approximation theory for functions of one or several variables [17]-[21], in the theory of quadrature formulas [6,22] and wavelet transformations [8]-[11], [23,24]. In the paper [25], an analog of the sampling theorem that employs the Hermite type interpolation was obtained.The authors of [26] succeeded in proving approximability for less smooth functions. However, they were forced to modify the operator (1) somewhat. In [26], it was established that if a uniformly continuous and bounded functions f on R belongs to the Dini-Lipchitz class and satisfies the condition f (x) = O(|x| −δ ) as x → ±∞ for some 2010 Mathematics Subject Classification. Primary 42A10, 41A30.

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“…In these papers band limited functions on homogeneous spaces (or more generally manifolds with bounded geometry) are reconstructed from irregularly spaced samples using Lagrangian splines or iterative algorithms. Sampling theorems on symmetric spaces have also been obtained; see [4,5,10,19]. In [10] the iterative reconstruction algorithm of [9] is made explicit for the hyperbolic plane.…”

Section: Introductionmentioning

confidence: 99%

“…In [19] a band-limited function on a noncompact symmetric space is recovered from samples of its convolution with certain bi-invariant distributions of compact support, using either abstractly defined frames or polyharmonic splines as the analog of the sinc function. Our work is perhaps closest in spirit to [4,5], in which a function is reconstructed from samples of its Radon transform.…”

Section: Introductionmentioning

confidence: 99%

A Reconstruction Theorem for Riemannian Symmetric Spaces of Noncompact Type

Stenzel

2009

J Fourier Anal Appl

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Let f be a rapidly decreasing radial function on a Riemannian symmetric space of noncompact type whose spherical Fourier transform has compact support. We prove a reconstruction theorem which recovers f from the values of an integral operator applied to f on a discrete subset. When G/K is of the complex type we prove a sampling formula recovering f from its own values on a discrete subset. We give explicit results for three dimensional hyperbolic space.

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On Sampling Formulas on Symmetric Spaces

Abstract: We give a sampling formula using the Radon transform along a maximal geodesic subspace of the Riemannian symmetric space. For the real hyperbolic space we can get a total sampling formula. To get this formula, we prepare a sampling formula for the sphere.

Cited by 9 publications

References 9 publications

Sampling in Paley-Wiener spaces on combinatorial graphs

Sampling in Paley-Wiener spaces on combinatorial graphs

On divergence of sinc-approximations everywhere on $(0,\pi)$

A Reconstruction Theorem for Riemannian Symmetric Spaces of Noncompact Type

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