Summary
Scaling functions that generate a multiresolution analysis (MRA) satisfy, among other conditions, the so-called «two-scale relation» (TSR). In this paper we discuss a number of properties that follow from the TSR alone, independently of any MRA: position of zeros (mainly for continuous scaling functions), existence theorems (using fixed point and eigenvalue arguments) and orthogonality relation between integer translates.
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Work supported by Fondazione Angelo Della Riccia.
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Antoine, J.P., Bagarello, F. Some analytical considerations on two-scale relations. Nuov Cim B 109, 871–890 (1994). https://doi.org/10.1007/BF02722465
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DOI: https://doi.org/10.1007/BF02722465