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Step refinable functions and orthogonal MRA on Vilenkin groups. (English) Zbl 1307.42035

The author approaches the problem of constructing refinable functions and multiresolution analysis (MRA) on the zero-dimensional locally compact Vilenkin groups. He provides necessary and sufficient conditions on a refinable step function under which this function generates an orthogonal MRA in the \(L_2(G)\) spaces on the Vilenkin group G. A special class of refinable step functions (defined with restrictions on the mask) that generates an orthogonal MRA on Vilenkin groups is characterized in terms of a support condition on the function Fourier transform. Moreover, the author provides a theorem that shows the sharpness of this result.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65T60 Numerical methods for wavelets
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
43A70 Analysis on specific locally compact and other abelian groups
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