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Wandering Vectors for Unitary Systems and Orthogonal Wavelets (Memoirs of the American Mathematical Society)

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The term "wandering vector" is from the theory of operators in Hilbert space;-- it means that the vector in question is transformed into a family of orthogonal vectors under a prescribed set of unitary operators. Since wavelets represent
orthogonal functions in the Hilbert space of square integrable functions on R, or on R^n for some n, it would only seem natural to merge the two concepts. But it takes originality to carry through the program: The book represents a success story in the application of operator theory to a current problem (in this case, wavelets) in classical analysis. What came out, for the theory of wavelets, was at first quite unexpected: You could have wavelets, it turned out, that are generated by fewer functions on R, or on R^n, than
had been predicted by the more conventional approach to wavelet theory. One generator is enough! The new wavelets turned out to be localized in frequency domain, and they gave rise to a variety of new trends in wavelet theory, the wavelet sets being just one of them, multiplicity, and the dimension function are others.