A probabilistic approach to scaling equations

Summary.

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.