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Open AccessArticle

Stability of Spline-Type Systems in the Abelian Case

by Darian Onchis 1,2,*,† and Simone Zappalà 1,†
1
Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
2
Department of Computer Science, West University of Timisoara, Bulevardul Vasile Pârvan 4, 300223 Timisoara, Romania
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2018, 10(1), 7; https://doi.org/10.3390/sym10010007
Received: 29 November 2017 / Revised: 20 December 2017 / Accepted: 21 December 2017 / Published: 27 December 2017
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. View Full-Text
Keywords: spline-type systems; stability; biorthogonal systems; multi-level schemes; constructive realizations spline-type systems; stability; biorthogonal systems; multi-level schemes; constructive realizations
MDPI and ACS Style

Onchis, D.; Zappalà, S. Stability of Spline-Type Systems in the Abelian Case. Symmetry 2018, 10, 7.

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