Measures in wavelet decompositions

Abstract

In applications, choices of orthonormal bases in Hilbert space H may come about from the simultaneous diagonalization of some specific abelian algebra of operators. This is the approach of quantum theory as suggested by John von Neumann; but as it turns out, much more recent constructions of bases in wavelet theory, and in dynamical systems, also fit into this scheme. However, in these modern applications, the basis typically comes first, and the abelian algebra might not even be made explicit. It was noticed recently that there is a certain finite set of non-commuting operators Fi, first introduced by engineers in signal processing, which helps to clarify this connection, and at the same time throws light on decomposition possibilities for wavelet packets used in pyramid algorithms. There are three interrelated components to this: an orthonormal basis, an abelian algebra, and a projection-valued measure. While the operators Fi were originally intended for quadrature mirror filters of signals, recent papers have shown that they are ubiquitous in a variety of modern wavelet constructions, and in particular in the selection of wavelet packets from libraries of bases. These are constructions which make a selection of a basis with the best frequency concentration in signal or data-compression problems. While the algebra A generated by the Fi -system is non-abelian, and goes under the name "Cuntz algebra" in C*-algebra theory, each of its representations contains a canonical maximal abelian subalgebra, i.e., the subalgebra is some C( X ) for a Gelfand space X. A given representation of A , restricted to C( X ) , naturally induces a projection-valued measure on X, and each vector in H induces a scalar-valued measure on X. We develop this construction in the general context with a view to wavelet applications, and we show that the measures that had been studied earlier for a very restrictive class of Fi -systems (i.e., the Lemarié-Meyer quadrature mirror filters) in the theory of wavelet packets are special cases of this. Moreover, we prove a structure theorem for certain classes of induced scalar measures. In the applications, X may be the unit interval, or a Cantor set; or it may be an affine fractal, or even one of the more general iteration limits involving iterated function systems consisting of conformal maps. © 2004 Elsevier Inc. All rights reserved.