Perfect-Reconstruction Filtering With Unitary Operators and Projections*

Perfect-reconstruction filter banks, wavelets, and multiresolution analysis have generated many new results in harmonic analysis and function theory. Traditionally, these techniques are based on families of convolutions and decimation/interpolation operations. When viewed as linear-algebraic operations, those building blocks can be generalized to unitary operators and related projections. We have developed a general approach to perfect reconstruction and multiresolution analysis that uses polynomial functions of unitary operators and closely related projection operations. This general framework is powerful in that it easily captures much of the known theory as special cases and gives a simple mechanism for constructing arbitrary perfect reconstruction families of operators. Using the general framework, we derive necessary and sufficient conditions for the perfect reconstruction property, even for higher-dimensional nonseparable sampling lattices and operators. The necessary conditions can be interpreted in terms of the spectral mapping theorem and, although explicitly stated, are not directly computable. *Research partially supported by AFOSR contract F49620-93-I-0226. LINEAR ALGEBRA AND ITS APPLICATIONS 208/209:97-133 (1994) 97 0 Elsevier Science Inc., 1994 655 Avenue of the Americas, New York, NY 10010 0024-3795/94/$7.00