Filter banks and perfect reconstruction in finite dimensional spaces

We consider the problem of developing filter banks with the perfect reconstruction property for finite dimensional signals. We are motivated by the discrete, finite dimensional character of digital signals and images, which naturally leads to the study of the discrete counterpart of multiresolution analysis and wavelet series expansions in infinite dimensional spaces such as L₂ and l₂. In finite dimensional spaces, all computations can be performed using finite matrix operations. The discrete Fourier transform (DFT) is the natural tool for the harmonic analysis in such spaces, in which the circular convolution operation plays a vital role. There has also been interest in this problem by other authors. However, our approach is distinct: in a sense, it is simpler and more independent of the well-known theory in L₂