(The) stability of integer translations of a function and its application to MRA construction

Abstract

This thesis is devoted to the study of the stability of integer translates of a function and sampling functions and its applications to multiresolution analysis (MRA) construction. First, we review the concept of MRA and give some important results on MRA. We also study the stability of the integer translates of the characteristic functions of one interval and the functions whose Fourier transforms are the characteristic functions of one interval. Next, we improve the well-known Cohen`s theorem using the concept of ``congruence to [-π,π] modulo 2π" for the refinable stable functions by modifying the set [-π,π]. We characterize the orthonormal scaling functions of length of support less than 5 by applying the Cohen`s cycle theorem. We also show that the preservation of stability under the convolution is related with the zero set of the Fourier transform of inducing stable function and the preservation of stability for compactly supported refinable function is related with the zero set of its mask. Finally, we give a necessary and sufficient condition for a refinable sampling functions which connects the sampling function and stable function. We also give the dual functions of sampling functions, a method of constructing the sampling function in $V_1$ instead of $V_0$, and some examples.