Some topics on refinable functions

Abstract

This thesis consists of two different parts, related by the common theme of refinability. One part is to explore some results of paper [1]. We will use some results of the paper to prove that the set S of all refinable functions is nowhere dense in L_2(IR) and there is no upper bound for the set {d(f; S):f in L_2(IR)} of distances. A concrete set of fully refinable functions will be given. Another part is based on paper [2]. The matrix-valued FejA'er-Riesz lemma A(z) = B(1/z)^TB(z), for which A(z) is positive semi-definite on the torus, will be studied and applied to a construction of refinable local orthonormal generators from given local generators with linear independent shifts. A construction of multiwavelet masks from refinement masks by the use of paraunitary matrices will be introduced as the end of this thesis. Some programs written in Maple V for these constructions are attached.[1] G. Strang and D.X. Zhou, The limits of refinable functions, Trans. Amer. Math. Soc., vol.353 n5, pp.1971-1984, 2001. [2] C.K. Chui, W. He, J. Stockler and Q.Y. Sun, Compactly supported tight affine frames with integer dilations and maximum vanishing moments, Adv. Comp. Math., vol.18, pp.297-327,2003.