Robertson Type Theorems for Frames

Abstract

We extend Robertson's theorem to apply to frames generated by the action of a discrete, countable abelian unitary group. Within this setup we use Stone's theorem and the theory of spectral multiplicity to analyze wandering frame collections. Motivated by wavelet theory, we explicitly apply our results to the action of the integers given by translations on $\ltwo$. This yields a new functional analytic method of constructing a wavelet from a multiresolution analysis.