Slanted matrices, Banach frames, and sampling

Abstract

In this paper we present a rare combination of abstract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sampling 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 matrices.

Stability of localized operators

“…This result for the special weights w(x) = (1 + |x|) s , with s > (d + 1) 2 and with s > 0, is established in [3] and [65], respectively.…”

“…The equivalence of q -stability of an infinite matrix A = (a(j, j )) j,j ∈Z with k∈Z sup j ∈Z |a(j, j +k)|(1+ |k|) s < ∞ is established in [3] for s > 4 and later improved in [65] for s > 0.…”

“…The slanted matrices appear in wavelet theory, signal processing and sampling theory [3,11,16,45], and also occur in the K-theory of operator algebras [71]. Note that…”

“…(1.2) For the convolution operator C a associated with a summable sequence a = (a(j )) j ∈Z , it is known that C a has p -stability for some 1 p ∞ if and only if a(ξ ) = 0 for all ξ ∈ R, (1. 3) whereâ(ξ ) := j ∈Z a(j )e −ij ξ (cf. [2,43]).…”

“…Inspired by the commutator technique developed in [55] and norm equivalence technique for a finite-dimensional space in [3], we will give a new proof of Theorem 1.1 in this paper without using the characterization (1.3). More importantly we can extend the equivalence for p -stability in Theorem 1.1 to various localized operators of non-convolution type, that arise in the study of spline approximation [26,28], wavelet and affine frames [20,42], Gabor frame and non-uniform sampling [7,35,36,60], and pseudo-differential operators [13,14,29,34,38,40,55,56,[66][67][68].…”