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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.

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