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 recent work 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 previous work by the authors, 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.
(*) Original article of March 2003, revised March 2004.