Nonstationary tight wavelet frames, I: Bounded intervals

Abstract

The notion of tight (wavelet) frames could be viewed as a generalization of orthonormal wavelets. By allowing redundancy, we gain the necessary flexibility to achieve such properties as “symmetry” for compactly supported wavelets and, more importantly, to be able to extend the classical theory of spline functions with arbitrary knots to a new theory of spline-wavelets that possess such important properties as local support and vanishing moments of order up to the same order of the associated B-splines. This paper is devoted to develop the mathematical foundation of a general theory of such tight frames of nonstationary wavelets on a bounded interval, with spline-wavelets on nested knot sequences of arbitrary non-degenerate knots, having an appropriate number of knots stacked at the end-points, as canonical examples. In a forthcoming paper under preparation, we develop a parallel theory for the study of nonstationary tight frames on an unbounded interval, and particularly the real line, which precisely generalizes the recent work [Appl. Comput. Harmon. Anal. 13 (2002) 224–262; Appl. Comput. Harmon. Anal. 14 (2003) 1–46] from the shift-invariance setting to a general nonstationary theory. In this regard, it is important to point out that, in contrast to orthonormal wavelets, tight frames on a bounded interval, even for the stationary setting in general, cannot be easily constructed simply by using the tight frame generators for the real line in [Appl. Comput. Harmon. Anal. 13 (2002) 224–262; Appl. Comput. Harmon. Anal. 14 (2003) 1–46] and introducing certain appropriate boundary functions. In other words, the general theories for tight frames on bounded and unbounded intervals are somewhat different, and the results in this paper cannot be easily derived from those of our forthcoming paper. The intent of this paper and the forthcoming one is to build a mathematical foundation for further future research in this direction. There are certainly many interesting unanswered questions, including those concerning minimum support, minimum cardinality of frame elements on each level, “symmetry,” and order of approximation of truncated frame series. In addition, generalization of our development to sibling frames already encounters the obstacle of achieving Bessel bounds to assure the frame structure.