B-Spline Quarklets and Biorthogonal Multiwavelets

TL;DR

This paper introduces B-spline quarklets within biorthogonal multiwavelet theory, demonstrating their properties, reconstruction capabilities, and potential for non-orthogonal decompositions of L2(R).

Contribution

It establishes the theoretical framework for B-spline quarklets, including their multiresolution analysis properties, duals, and reconstruction matrices, advancing wavelet theory.

Findings

Quarklets fit into biorthogonal multiwavelet theory.

Existence of dual quarks and quarklets with compact support.

Quarks and quarklets enable non-orthogonal decompositions of L2(R).

Abstract

We show that B-spline quarks and the associated quarklets fit into the theory of biorthogonal multiwavelets. Quark vectors are used to define sequences of subspaces $ V_{p,j} $ of $ L_{2}(\mathbb{R}) $ which fulfill almost all conditions of a multiresolution analysis. Under some special conditions on the parameters they even satisfy all those properties. Moreover we prove that quarks and quarklets possess modulation matrices which fulfill the perfect reconstruction condition. Furthermore we show the existence of generalized dual quarks and quarklets which are known to be at least compactly supported tempered distributions from $\mathcal{S}'(\mathbb{R})$. Finally we also verify that quarks and quarklets can be used to define sequences of subspaces $ W_{p,j} $ of $ L_{2}(\mathbb{R}) $ that yield non-orthogonal decompositions of $ L_{2}(\mathbb{R}) $.