aequationes mathematicae

, Volume 58, Issue 3, pp 242–259 | Cite as

A probabilistic approach to scaling equations

  • A. O. Pittenger
  • J. V. Ryff


In wavelet theory an equation of the form¶¶\( \varphi(x) = \sum\limits^d_{k = 0} c_k \varphi(2x - k), \quad -\infty \) < x < \(\infty \)(S)¶¶with real coefficients c k is called a scaling equation, and real-valued solutions of scaling equations with compact support play a central role in the construction of wavelets through the process of multiresolution analysis. In this paper we examine hypotheses on the constants and solutions of the scaling equation and use probabilistic techniques to obtain explicit representations of all solutions for the special case of d = 2. These techniques are also applied to establish uniqueness of the constants when d = 1 and reference is made to similar results for arbitrary finite d.

Keywords. Wavelets, scaling equations, martingales. 


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Copyright information

© Birkhäuser Verlag, Basel, 1999

Authors and Affiliations

  • A. O. Pittenger
    • 1
  • J. V. Ryff
    • 2
  1. 1.Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USAUSA
  2. 2.P.O. Box 12, South Harwich, Massachusetts 02661, USAUSA

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