The Time operator of wavelets

This paper establishes an interesting new and general connection between the wavelet theory of harmonic analysis and the Time operator theory of statistical physics. In particular, it will be shown that an arbitrary wavelet multiresolution analysis (MRA) defines a Time operator T whose age eigenspaces are the wavelet detail subspaces . Extension of this result to the continuous parameter case induces a new notion of continuous wavelet multiresolution analysis. The Time operator T incorporates and exhibits in a natural way all five fundamental properties of a wavelet multiresolution analysis.