Wavelets, frames and operator theory
by
David Larson
Texas A&M University

Orthonormal wavelets can be regarded as complete wandering vectors for a system of bilateral shifts acting on a separable infinite dimensional Hilbert space. We describe in detail a method of constructing new wavelets by interpolating between known pairs of wavelets, and more generally interpolating over finite families of wavelets, using Hilbert space operator techniques. These considerations lead naturally to some basic global results for orthonormal wavelets, Riesz wavelets and frame wavelets, that can tend to be surprising from a purely function-theoretic point of view. Included is the concept of a super wavelet (a wavelet for a super-space), which has potential applications to multiplexing problems, and we give some new examples of superwavelets. These considerations also yield alternate derivations of some well-known function-theoretic results. Similar techniques can be applied to the windowed Fourier transforms found in Gabor Theory, or Weyl-Heisenberg Frame Theory, and we obtain some global density and connectivity results for these classes.

Date received: June 10, 2002