Wavelets for analyzing scattered data: An unbounded operator approach

The objective of this paper is to provide a general framework for the construction of wavelets that are generated by translates of a given slowly growing function where the translates are defined by a sequence of arbitrarily spaced points. Our approach is operator theoretic in nature and relies only on specific properties of the function (e.g., the order of the singularity of its Fourier transform at the origin) and not on a detailed knowledge of the function itself. The methods used here not only provide a unifying thread for several known results but allow for many new results as well. For example, the first results on stability of certain wavelet bases associated with scattered data on the line are obtained here (the only assumption on the scattered data is boundedness of the global mesh ratio), Many radial basis functions are unbounded on the line and this fact leads to our investigation of certain classes of unbounded operators including their domain identification and their closed extensions. An important result of this study is an exact identification of the L(2)(R) spaces generated by linear combinations of shifts of certain unbounded radial basis functions. (C) 1996 Academic Press, Inc.