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Stability of Spline-Type Systems in the Abelian Case
Darian Onchis1,2, Simone Zappalà1
1Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria.
Symmetry
|February 3, 2018
Summary
This study characterizes the stability of spline-type spaces using operator boundedness. Linear independence ensures synthesis operator stability, while uniform boundedness of projection operators characterizes stable sequences.
Area of Science:
- Functional Analysis
- Harmonic Analysis
- Approximation Theory
Background:
- The stability of translation-invariant spaces is crucial in harmonic analysis and approximation theory.
- Characterizing stability often involves analyzing the properties of synthesis and projection operators.
- Spline-type spaces are fundamental in numerical analysis and signal processing.
Purpose of the Study:
- To characterize the stability of translation-invariant spaces of distributions over locally compact groups.
- To provide a characterization of the stability of spline-type spaces in terms of operator boundedness.
- To investigate the stability of sequences of spline-type spaces using multiresolution analysis concepts.
Main Methods:
- The study defines stability as the boundedness of synthesis and projection operators.
- Linear independence of basis functions is used to characterize lower boundedness of the synthesis operator.
- The Lax equivalence principle for discretization schemes inspires the approach to uniform boundedness of projection operators.
Main Results:
- A characterization of the stability of spline-type spaces is established via linear independence, linking it to the lower boundedness of the synthesis operator.
- A constructive method for realizing biorthogonal systems is presented, derived from the proof of Theorem 2.
- Theorem 3 characterizes stable sequences of stable spline-type spaces through the uniform boundedness of projection operators.
Conclusions:
- The stability of translation-invariant spaces is directly related to the boundedness of associated operators.
- Linear independence serves as a key condition for the stability of spline-type spaces.
- The findings provide a comprehensive framework for understanding the stability of sequences of spline-type spaces.
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