The Characters of Quarternary Scaling Functions and Their Generators with Twelve-scale Dilation Factor

In this paper, the notion of matrix-valued multiresolution analysis is introduced. A procedure for designing biorthogonal matrix-valued quarternary wave wrap functions in is presented and their characters are researched by virtue of time-frequency analysis method, matrix theory and operator theory. Three biorthogonality formulas concerning the wave wrap functions are obtained. Finally, new Riesz bases of four dimensional matrix-valued function space are derived by designing a series of subspaces of biorthogonal matrix-valued wave wrap functions.