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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A continued fraction analysis of periodic wavelet coefficients
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by Joel Glenn
Proc. Amer. Math. Soc. 132 (2004), 1367-1375
DOI: https://doi.org/10.1090/S0002-9939-03-07064-3
Published electronically: December 22, 2003
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Abstract:

We define and prove the existence of crossings of wavelet coefficients translated by integer multiples of the numerator of a continued fraction convergent of the ratio of the sampling interval to the period of the wavelet coefficients. Crossings are found to be translation invariant $\pm 1$. Intervals between crossings are analyzed for wavelets with $n$ vanishing moments. These wavelets act as multiscale differential operators. These crossings reveal different locations in the period where there is equality in the $n$th derivative of an averaging of the signal. These results will be employed in the estimation of frequency components in future publications.
References
  • C. Burrus, R. Gopinah, and H. Guo, Introduction to Wavelets and Wavelet Transforms: A Primer, Prentice-Hall, Inc., Upper Saddle River, New Jersey, 1998, pp. 190-195.
  • William J. LeVeque, Fundamentals of number theory, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977. MR 0480290
  • Stéphane Mallat, A wavelet tour of signal processing, Academic Press, Inc., San Diego, CA, 1998. MR 1614527
  • A. Ya. Khintchine, Continued fractions, P. Noordhoff Ltd., Groningen, 1963. Translated by Peter Wynn. MR 0161834
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Bibliographic Information
  • Joel Glenn
  • Affiliation: Department of Mathematics and Computer Science, Colorado College, Colorado Springs, Colorado 80903
  • Email: jglenn@coloradocollege.edu
  • Received by editor(s): February 19, 2002
  • Received by editor(s) in revised form: September 26, 2002
  • Published electronically: December 22, 2003
  • Communicated by: David R. Larson
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 1367-1375
  • MSC (2000): Primary 42C40, 65T60; Secondary 11A55, 40A15
  • DOI: https://doi.org/10.1090/S0002-9939-03-07064-3
  • MathSciNet review: 2053341