Wiener's lemma for infinite matrices with polynomial off-diagonal decay

Sun, Qiyu

doi:10.1016/j.crma.2005.03.002

Cited by 52 publications

(88 citation statements)

References 17 publications

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“…There are many different approaches to establish Wiener's lemma for infinite matrices in the Schur class A p,w (X, ρ, µ). Here are three of them: (i) the indirect approach, such as using Gelfand's technique to estimate spectral radius ρ p,w (A) ( [4,17,18,23,24]); (ii) the semi-direct approach, such as the bootstrap argument ( [26]); (iii) the direct approach, such as the direct estimate of A n A p,w in Theorem 3.1 (see [36] for the baby version of that approach). Each approach has its advantages and disadvantages.…”

Section: Wiener's Lemma For Infinite Matrices In the Schur Classmentioning

confidence: 99%

Wiener’s lemma for infinite matrices

Sun

2007

Trans. Amer. Math. Soc.

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129

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Abstract. The classical Wiener lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener lemma, with an equivalent formulation suitable for our generalization involving commutative algebra of infinite matricesIn the study of spline approximation, (diffusion) wavelets and affine frames, Gabor frames on non-uniform grid, and non-uniform sampling and reconstruction, the associated algebras of infinite matrices are extremely non-commutative, but we expect those noncommutative algebras to have a similar property to Wiener's lemma for the commutative algebra W. In this paper, we consider two non-commutative algebras of infinite matrices, the Schur class and the Sjöstrand class, and establish Wiener's lemmas for those matrix algebras.

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Section: Wiener's Lemma For Infinite Matrices In the Schur Classmentioning

confidence: 99%

Wiener’s lemma for infinite matrices

Sun

2007

Trans. Amer. Math. Soc.

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129

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“…The third property in Lemma 4.10 below is usually known as the Wiener's lemma, see, for instance, [4,24,31,32,35,56,57] and references therein for its recent development and various applications. Lemma 4.10.…”

Section: Proof Of Theorem 41mentioning

confidence: 99%

Frames in spaces with finite rate of innovation

Sun

2006

Adv Comput Math

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Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space V q ( , ) modeling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wideband communication. In particular, the space V q ( , ) is generated by a family of well-localized molecules of similar size located on a relatively separated set using q coefficients, and hence is locally finitely generated. Moreover that space V q ( , ) includes finitely generated shift-invariant spaces, spaces of non-uniform splines, and the twisted shift-invariant space in Gabor (Wilson) system as its special cases. Use the well-localization property of the generator , we show that if the generator is a frame for the space V 2 ( , ) and has polynomial (sub-exponential) decay, then its canonical dual (tight) frame has the same polynomial (sub-exponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator for the space V q ( , ) with q = 2, and of the polynomial (sub-exponential) decay property of the mask associated with a refinable function that has polynomial (sub-exponential) decay.Keywords frame · Banach frame · localized frame · signals with finite rate of innovation · space of homogenous type · matrix algebra · refinable function · wavelets Mathematics Subject Classifications (2000) Primary 42C40 · Secondary 47B37 · 46C07

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“…This inspired us to consider a numerically implementable reconstruction formula using partial sampling data and provide an estimate to the reconstruction error, see Theorem 2 for details. In order to prove Theorem 2 we need Wiener's lemma for infinite matrices [12,15,20,22]. Therefore for the rest of the paper we assume that…”

Section: Partial Reconstructionmentioning

confidence: 99%

On a class of non-uniform average sampling expansions and partial reconstruction in subspaces of L 2(ℝ)

Atreas

2011

Adv Comput Math

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Let φ be a function in the Wiener amalgam space W ∞ (L 1 ) with a non-vanishing property in a neighborhood of the origin for its Fourier transform φ, τ = {τ n } n∈Z be a sampling set on R and V τ φ be a closed subspace of L 2 (R) containing all linear combinations of τ -translates of φ. In this paper we prove that every function f ∈ V τ φ is uniquely determined by and stably reconstructed from the sample set L τAs our reconstruction formula involves evaluating the inverse of an infinite matrix we consider a partial reconstruction formula suitable for numerical implementation. Under an additional assumption on the decay rate of φ we provide an estimate to the corresponding error.

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Wiener's lemma for infinite matrices with polynomial off-diagonal decay

Cited by 52 publications

References 17 publications

Wiener’s lemma for infinite matrices

Wiener’s lemma for infinite matrices

Frames in spaces with finite rate of innovation

On a class of non-uniform average sampling expansions and partial reconstruction in subspaces of L 2(ℝ)

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