Some stability properties of wavelets and scaling functions

Abstract

The property of continuity of an arbitrary scaling function is known to be unstable with respect to the coefficients in the associated dilation equation. That is, if f is a continuous function which is a solution of the dilation equation \(f(x) = \sum _{k = 0}^N c_k f(2x - k)\) then a dilation equation with slightly perturbed coefficients {cO,…, cN } need not have a continuous solution. The convergence of the Cascade Algorithm, an iterative method for solving a dilation equation, is likewise unstable in general. This paper establishes a condition under which stability does occur: both continuity and uniform convergence of the Cascade Algorithm are shown to be stable for those initial choices of coefficients {cO,…, cN } such that the integer translates of the scaling function f are l ∞ linearly independent. In particular, this applies to those scaling functions which can be used to construct orthogonal or biorthogonal wavelets. We show by example that this l ∞ linear independence condition is not necessary for stability to occur.