Two-direction refinable functions and two-direction wavelets with high approximation order and regularity

Abstract

The concept of two-direction refinable functions and two-direction wavelets is introduced. We investigate the existence of distributional(or L 2-stable) solutions of the two-direction refinement equation: $$\phi (x) = \sum\limits_k {p_k^ + } \phi (mx - k) + \sum\limits_k {p_k^ - } \phi (k - mx)$$ , where m ⩾ 2 is an integer. Based on the positive mask {p k + } and negative mask {p k − }, the conditions that guarantee the above equation has compactly distributional solutions or L 2-stable solutions are established. Furthermore, the condition that the L 2-stable solution of the above equation can generate a two-direction MRA is given. The support interval of ϕ(x) is discussed amply. The definition of orthogonal two-direction refinable function and orthogonal two-direction wavelets is presented, and the orthogonality criteria for two-direction refinable functions are established. An algorithm for constructing orthogonal two-direction refinable functions and their two-direction wavelets is presented. Another construction algorithm for two-direction L 2-refinable functions, which have nonnegative symbol masks and possess high approximation order and regularity, is presented. Finally, two construction examples are given.