Glasnik Matematicki, Vol. 46, No. 2 (2011), 415-431.

A FINITE-DIMENSIONAL APPROACH TO WAVELET SYSTEMS ON THE CIRCLE

Brody Dylan Johnson

Abstract.   Motivated by recent developments in the study of finite-dimensional frames, this work develops an independent theory of finite-dimensional wavelet systems on the circle. Using natural translation and dilation operators, trigonometric polynomial, orthonormal scaling functions are constructed which give rise to finite-dimensional multiresolution analyses and, consequently, orthonormal wavelet systems. It is shown that the finite-dimensional systems so constructed can lead to arbitrarily close approximation of square-integrable functions on the circle. Departures from the existing theory of periodic wavelets are encountered, e.g., the finite-dimensional equivalent of the Smith-Barnwell equation describes both a necessary and sufficient condition on a candidate low-pass filter for the existence of an orthonormal scaling function. Moreover, this finite-dimensional framework allows for a natural analog to the Shannon wavelet, in contrast to the classical periodic wavelets.

2000 Mathematics Subject Classification.   42C15, 65T60.

Key words and phrases.   Wavelet, circle, periodic wavelet, multiresolution analysis.

DOI: 10.3336/gm.46.2.11

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