Characterization of approximation order of multi-scaling functions via refinable super functions

Abstract

In this work, we derive via refinable super functions characterization of approximation order of multi-scaling functions both in time and in frequency domains. It is shown that approximation order is achieved if the linear operator defined as the difference of the down-sampled convolution matrix and a matrix associated with the super function used has a zero eigenvalue. The left eigenvectors associated with the zero eigenvalue define the combinations of scaling functions that produce the desired refinable super function. In the frequency domain, the approximation order condition is expressed in terms of the refinement masks of the multi-scaling functions and the refinable super function. It shown that, implicit in this new characterization lies some well known results on approximation order. A matrix equality that equates the frequency characterization presented in this paper and Strang's well known characterization of accuracy is derived. It is shown that approximation order of multi-scaling functions can always be achieved by a refinable, compactly supported super function.