Quasi-Orthogonality and Quasi-Projections
M. Unser
Applied and Computational Harmonic Analysis, vol. 3, no. 3, pp. 201–214, July 1996.
Our main concern in this paper is the design of simplified filtering procedures for the quasi-optimal approximation of functions in subspaces of L 2 generated from the translates of a function φ(x). Examples of signal representations that fall into this framework are Schoenberg's polynomial splines of degree n, and the various multiresolution spaces associated with the wavelet transform. After a brief review of the relation between the order of approximation of the representation and the concept of quasi-interpolation (Strang-Fix conditions), we investigate the implication of these conditions on the various basis functions and their duals (vanishing moment and quasi-interpolation properties). We then introduce the notion of quasi-duality and show how to construct quasi-orthogonal and quasi-dual basis functions that are much shorter than their exact counterparts. We also consider the corresponding quasi-orthogonal projection operator at sampling step h and derive asymptotic error formulas and bounds that are essentially the same as those associated with the exact least-squares solution. Finally, we use the idea of a perfect reproduction of polynomials of degree n to construct short kernel quasi-deconvolution filters that provide a well-behaved approximation of an oblique projection operator.
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