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

We derive, via refinable super functions, the characterization of the approximation order of multi-scaling functions in both the time and frequency domains. It is shown that the approximation order is achieved if a 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 is shown that, implicit in this new characterization, there lie some well known results on approximation order. A matrix equality is derived that equates the presented frequency characterization and Strang´s well known characterization of accuracy. It is shown that the approximation order of multi-scaling functions can always be achieved by a refinable, compactly supported super function.