A measure theoretic approach to the construction of scaling functions for wavelets

Abstract A multiresolution analysis is a tool used in the construction of orthogonal wavelets. The dilation equation is an equation that arises naturally when using an MRA to construct a wavelet basis. One way to understand the dilation equation, and its solution, the scaling function, is through a measure theoretic approach. By constructing a solution to the signed measure dilation equation, we give a new way of approximating the scaling function by dyadic step functions. We also give a method of controlling the support in the two-dimensional case.