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Sobolev characterization of solutions of dilation equations. (English) Zbl 0761.42014
A technique of positive operators is applied to the study of the regularity (or smoothness) of the solutions of the two-scale (or “refinement” or “dilation”) equations in the formulation of a multiresolution analysis. The sharp limit of the Sobolev exponent of the solution is given in terms of the spectral radius of a corresponding finite-dimensional positive operator. In addition, tools are given for yielding explicit upper and lower bounds of the exponent. Two graphs are included to demonstrate the sharpness of the results.

MSC:
42C15 General harmonic expansions, frames
15B48 Positive matrices and their generalizations; cones of matrices
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