SLANTED MATRICES, BANACH FRAMES, AND SAMPLING

By Akram Aldroubi, A. Baskakov, Ilya A. Krishtal. 2007

85 citations recorded by the source corpus.

In this paper we present a rare combination of ab- stract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sam- pling problems. We show that for a large class of slanted matrices boundedness below of the corresponding operator in ' p for some p implies boundedness below in ' p for all p. We use the established result to enrich our understanding of Banach frames and obtain new results for irregular sampling problems. We also present a version of a non-commutative Wiener's lemma for slanted matri- ces.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8d60d4e04b1f23ee8d38dd0519e130ca705a0765

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Citations in the corpus

Dynamical sampling (2014, methodology): “The methods that we develop for studying the dynamical sampling problem are related to methods in spectral theory, operator algebras, and frame theory [2, 10, 13, 15, 18, 20, 21, 22, 45].”
Iterative actions of normal operators (2016, background): “For us, the motivation to study the iterative actions of normal operators comes from sampling theory and related topics [2, 9, 21, 29, 31, 5, 6, 18, 37, 38, 16, 15].”
Stability of Localized Operators (2008, methodology): “Preprint submitted to Journal of Functional Analysis 11 November 2008”
Dynamical Sampling and Systems from Iterative Actions of Operators (2016, background): “The dynamical sampling problem has connections and applications to other areas of mathematics including, Banach algebras, C∗-algebras, spectral theory of normal operators, and frame theory [2, 8, 10, 13, 14, 15, 26, 32].”
Wiener’s Lemma for Infinite Matrices II (2007, methodology): “te Aq-weight for any 1 ≤ q < ∞, we have the following corollary of Theorem 4.1. Similar result about ℓq-stability for diﬀerent exponents q ∈ [1,∞] is established by Aldroubi, Baskakov and Krishtal [1] for inﬁnite matrices in the Gohberg-Baskakov-Sj¨ostrand class C 1,pα(Z d,Zd) with α > (d+1)2,”
Deformation of Gabor systems (2013, methodology): “It combines methods from the theory of localized frames [22, 25], the stability of operators on `-spaces [1, 36] and weak limit techniques in the style of Beurling [7].”