Orthogonal two-direction multiscaling functions

Abstract

The concept of a two-direction multiscaling functions is introduced. We investigate the existence of solutions of the two-direction matrix refinable equation $\Phi (x) = \sum\limits_{k \in \mathbb{Z}} {P_k^ + \Phi (2x - k) + } \sum\limits_{k \in \mathbb{Z}} {P_k^ - \Phi (k - 2x)} ,$ where r × r matrices {P_k ⁺} and {P_k ⁻} are called the positive-direction and negative-direction masks, respectively. Necessary and sufficient conditions that the above two-direction matrix refinable equation has a compactly supported distributional solution are established. The definition of orthogonal two-direction multiscaling function is presented, and the orthogonality criteria for two-direction multiscaling function is established. An algorithm for constructing a class of two-direction multiscaling functions is obtained. In addition, the relation of both orthogonal two-direction multiscaling function and orthogonal multiscaling function is discussed. Finally, construction examples are given.

Keywords

two-direction multiscaling function / positive-direction mask / negative-direction mask / two-direction matrix refinable equation / 42C15 / 94A12

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