Approximation order of two-direction multiscaling functions (Q2293669)

A matrix $A$ satisfies condition E if it has a simple eigenvalue of 1, and all other eigenvalues are smaller than 1 in absolute value. Condition E for two-direction multiscaling functions $\phi$ is necessary for the stability of the multiresolution approximation produced by $\phi$ or for the existence of $\phi$ in the matrix refinement equation of $\phi$. Condition E for $\Phi(x) := [\phi(x);\phi(-x)]^{T}$ is used as an equivalent condition for condition E for $\phi$. In this article the author obtains a simple and efficient criterion for condition E for $\phi$, criteria for the basic regularity conditions for $\phi$ as well as a criterion for approximation order of $\phi$. The author also gives examples for illustrating the general theory.