Dilation Equations with Exponential Decay Coefficients

Dilation equations with finitely many nonzero coefficients have been discussed by many people, since those equations are related to compactly supported wavelets, which constitute an important family of wavelets. However, some properties required in applications are not compatible with the compactness of the support; e.g., it is impossible to find an orthonormal and compactly supported scaling function with the sampling property except for a trivial case. Based on this fact, Xia and Zhang constructed a cardinal and orthonormal scaling function with exponential decay coefficients. It seems such a class of wavelets is interesting. We shall investigate that in this paper and in particular study the relationship between the exponential decay property of a scaling function and that of the corresponding coefficients.