Abstract
We explore compactly supported scaling functions of wavelet theory by means of classical umbral calculus as reformulated by Rota and Taylor. We set a theory of orthonormal scaling umbra which leads to a very simple and elementary proof of Lawton's theorem for umbrae. When umbrae come from a wavelet setting, we recover the usual Lawton condition for the orthonormality of the integer translates of a scaling function.
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Saliani, S., Senato, D. Compactly Supported Wavelets Through the Classical Umbral Calculus. J Fourier Anal Appl 12, 27–36 (2006). https://doi.org/10.1007/s00041-005-4085-y
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DOI: https://doi.org/10.1007/s00041-005-4085-y