Glasnik Matematicki, Vol. 46, No. 2 (2011), 415431.
A FINITEDIMENSIONAL APPROACH TO WAVELET SYSTEMS ON THE CIRCLE
Brody Dylan Johnson
Department of Mathematics and Computer Science, Saint Louis University, St. Louis, MO 63103, USA
Abstract. 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 finitedimensional multiresolution analyses and, consequently, orthonormal wavelet systems. It is shown that the finitedimensional systems so constructed can lead to arbitrarily close approximation of squareintegrable functions on the circle. Departures from the existing theory of periodic wavelets are encountered, e.g., the finitedimensional equivalent of the SmithBarnwell equation describes both a necessary and sufficient condition on a candidate lowpass 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.
2000 Mathematics Subject Classification.
42C15, 65T60.
Key words and phrases. Wavelet, circle, periodic wavelet, multiresolution analysis.
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DOI: 10.3336/gm.46.2.11
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