Biorthogonal Wavelet Expansions

This paper is concerned with developing conditions on a given nite collection of compactly supported algebraically linearly independent reenable functions that insure the existence of biorthog-onal systems of reenable functions with similar properties. In particular we address the close connection of this issue with stationary subdivision schemes. 1 Introduction During the past few years the construction of multivariate wavelets has received considerable attention. It is quite apparent that multivariate wavelets with good localazition properties in frequency and spatial domains which constitute an orthonormal basis of L 2 (IR s) are hard to realize. On the other hand, it turns out that in many applications orthogonality is not really important whereas locality, in particular, compact support is very desirable. In this regard, the concept of biorthogonality seems to ooer more exibility in practical realizations while still preserving many of the advantages of orthonormality. So far, this concept has been carefully studied in the univariate case (see e.g. CDF]). In the multivariate case concrete results have been obtained only for certain special bivariate examples CS, CD], and since the start of this paper multivariate studies have appeared in LC, KV]. The point of view taken in this paper is, to avoid trying to relax assumptions on the initial system used for the construction of a biorthogonal system. Instead we will focus on locality of the initial