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

The concept of two-direction refinable functions and two-direction wavelets is introduced.We investigate the existence of distributional(or L2-stable) solutions of the two-direction refinement equation: φ(x)=∑p+kφ(mx-k)+∑p-kφ(k-mx) where m ≥ 2 is an integer. Based on the positive mask {pk+} and negative mask {p-k}, the conditions that guarantee the above equation has compactly distributional solutions or L2-stable solutions are established. Furthermore, the condition that the L2-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 L2-refinable functions, which have nonnegative symbol masks and possess high approximation order and regularity, is presented. Finally, two construction examples are given.

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: (?)(x)=(?)p_k^+(?)(mx-k)+(?)p_k^-(?)(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 (?)(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 construct- ing 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.