Some Remarks on Multiscale Transformations, Stability and Biorthogonality

This paper is concerned with the concepts of stability and biorthogonality for a general framework of multiscale transformations. In particular, stability criteria are derived which do not make use of Fourier transform techniques but rather hinge upon classical Bernstein and Jackson estimates. Therefore they might be useful when dealing with possibly nonuniform discretizations or with bounded domains. x1. Introduction Let c be some string of data c k ; k 2 I, where I is some ((nite or possibly innnite) index set. These data could represent grey scale values of a digital image, statistical noisy data, or control points in some curve or surface representation , or approximate solutions of some discretized operator equation. The common ground for these rather diierent interpretations is that these data could be viewed as coeecients of some expansion f = X k2I c k ' k ; (1:1) where the ' k are (typically scalar-valued shape) functions deened on some domain (or manifold) (which is topologically equivalent to some bounded or unbounded domain) in IR s. As a familiar simple example one could take ' k as B-splines relative to some knot sequence in an interval. When each c k is a point in IR 3 say, f represents a space curve. The c k then already convey explicit geometrical information on the curve or, more precisely, on the location of the points f(x); x 2. It is well known that this kind of information can Curves and Surfaces II 1 P.