QUASI-ORTHOGONALITY AND QUASI-PROJECTIONS

Our main concern in this paper is the design of simplified filtering p rocedures for the quasi-optimal approximation of functions in subspace s of L(2) generated from the translates of a function phi(x). Examples of signal representations that fall into this framework are Schoenber g's polynomial splines of degree n, and the various multiresolution sp aces associated with the wavelet transform. After a brief review of th e relation between the order of approximation of the representation an d the concept of quasi-interpolation (Strang-Fix conditions), we inves tigate the implication of these conditions on the various basis functi ons and their duals (vanishing moment and quasi-interpolation properti es). We then introduce the notion of quasi-duality and show how to con struct quasiorthogonal and quasi-dual basis functions that are much sh orter than their exact counterparts. We also consider the correspondin g quasi-orthogonal projection operator at sampling step h and derive a symptotic error formulas and bounds that are essentially the same as t hose associated with the exact least-squares solution. Finally, we use the idea of a perfect reproduction of polynomials of degree n to cons truct short kernel quasi-deconvolution filters that provide a well-beh aved approximation of an oblique projection operator.