Step Refinable Functions and Orthogonal MRA on Vilenkin Groups |
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Authors: | Sergey F Lukomskii |
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Institution: | 1. Saratov, Russia
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Abstract: | We find necessary and sufficient conditions on refinable step function under which this function generates an orthogonal MRA in the $L_{2}(\mathfrak{G})$ -spaces on Vilenkin group $\mathfrak{G}$ . We consider a class of refinable step functions for which the mask m 0(χ) is constant on cosets $\mathfrak{G}_{-1}^{\bot}\chi$ and its modulus |m 0(χ)| has two values only: 0 and 1. We prove that any refinable step function φ from this class that generates an orthogonal MRA on Vilenkin group $\mathfrak{G}$ has Fourier transform with condition $\operatorname{supp}\hat{\varphi}(\chi)\subset\mathfrak{G}_{p-2}^{\bot}$ . We show the sharpness of this result, too. |
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