A Finite-dimensional Approach to Wavelet Systems on the Circle

Motivated by recent developments in the study of finitedimensional frames, this work develops an independent theory of finitedimensional 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 finitedimensional framework allows for a natural analog to the Shannon wavelet, in contrast to the classical periodic wavelets.