Stability of Spline-Type Systems in the Abelian Case

Abstract

In this paper, the stability of translation-invariant spaces of distributions over locally compact groups is stated as boundedness of synthesis and projection operators. At first, a characterization of the stability of spline-type spaces is given, in the standard sense of the stability for shift-invariant spaces, that is, linear independence characterizes lower boundedness of the synthesis operator in Banach spaces of distributions. The constructive nature of the proof for Theorem 2 enabled us to constructively realize the biorthogonal system of a given one. Then, inspired by the multiresolution analysis and the Lax equivalence for general discretization schemes, we approached the stability of a sequence of spline-type spaces as uniform boundedness of projection operators. Through Theorem 3, we characterize stable sequences of stable spline-type spaces.

Numerical stability of spline-based Gabor-like systems

“…For ST spaces, we have characterized the boundedness from below and the injectivity of the synthesis operator as linear independence of the Fourier transforms of the atoms on the orthogonal subgroup, see [17, Theorem 7] H⊥≔0.1em{x^∈G^:∀x∈H,0.3em〈x,x^〉=1}…”

Realizable algorithm for approximating Hilbert–Schmidt operators via Gabor multipliers

“…Theorem 1. [18] Let G be a LCA group, H be a discrete subgroup and Φ={ϕi}i=1R⊂L1(G) be a finite generating set of compactly supported distributions .…”