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B-spline quarklets and biorthogonal multiwavelets

Department of Mathematics and Computer Science, Philipps-Universität Marburg, Hans-Meerwein Str. 6, 35043 Marburg, Germany

E-mail Address: hovemann@mathematik.uni-marburg.de

Corresponding author.

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Anne Kopsch

Department of Mathematics and Computer Science, Philipps-Universität Marburg, Hans-Meerwein Str. 6, 35043 Marburg, Germany

E-mail Address: kopscha@mathematik.uni-marburg.de

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Thorsten Raasch

Department of Mathematics, University of Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany

E-mail Address: raasch@mathematik.uni-siegen.de

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, and

Dorian Vogel

Department of Mathematics and Computer Science, Philipps-Universität Marburg, Hans-Meerwein Str. 6, 35043 Marburg, Germany

E-mail Address: vogeldor@mathematik.uni-marburg.de

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https://doi.org/10.1142/S0219691323500297Cited by:2 (Source: Crossref)

Abstract

In this paper, 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 $L_{2} (ℝ)$ 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 $S^{'} (ℝ)$ . Finally, we also verify that quarks and quarklets can be used to define sequences of subspaces $W_{p, j}$ of $L_{2} (ℝ)$ that yield non-orthogonal decompositions of $L_{2} (ℝ)$ .

Keywords:

AMSC: 42C40, 41A15

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