Approximation order of two-direction multiscaling functions

Abstract

We investigate Condition E and the basic regularity conditions for the two-direction multiscaling functions $\boldsymbol {\phi }$, and then investigate approximation order of $\boldsymbol {\phi }$. By investigating a structure of approximation vectors, we are able to find simple and efficient criteria for Condition E, the basic regularity conditions, and the approximation order of $\boldsymbol {\phi }$. Examples for illustrating the general theory are given.