Proceedings of the Conference on Wavelet Analysis and Applications Applications.pdfVIP

Pro ceedings of the Conference on Wavelet Analysis and Applications Applications of shift-invariant space theory to some 2 d problems of multi-resolution analysis of L (R ) Hong Oh Kim and Jae Kun Lim 2 d Abstract. Given a frame multi-resolution analysis of L (R ) with a set of multiple scaling functions that is a quasi-stable generating set for V0 , we give some neccessary and sucient conditions under which the standard generating set for V 1 is also a quasi-stable generating set. We also give some equivalent conditions for W0 to admit a quasi-stable generating set by means of the re nement equation mask, under the condition that the standard generating sets for V0 and V 1 are quasi-stable generating sets. 1. Intro duction Ever since the intro duction of the multi-resolution analysis (MRA) by Mallat ([4]), its usefulness has b een recognized by many authors in numerous contexts. One of the myriad merits of MRA is that an MRA gives rise to a `wavelet de- comp osition. As the demands of many branches of mathematical and engineering sciences awaiting the applications of MRAs increase, the notion of MRA has b een generalized. In this pap er we consider the following generalized frame MRA of 2 d L (R ). 2 d Definition 1.1 . (fV g ;) is said to be an MRA of L (R ) if each V is a k k 2Z