Theory of the optimum interpolation approximation in a shift-invariant wavelet and scaling subspace

In the main part of this paper, we present a systematic discussion for the optimum interpolation approximation in a shift-invariant wavelet and/or scaling subspace. In this paper, we suppose that signals are expressed as linear combinations of a large number of base functions having unknown coefficients. Under this assumption, we consider a problem of approximating these linear combinations of higher degree by using a smaller number of sample values. Hence, error of approximation happens in most cases. The presented approximation minimizes various worst-case measures of approximation error at the same time among all the linear and the nonlinear approximations under the same conditions. The presented approximation is quite flexible in choosing the sampling interval. The presented approximation uses a finite number of sample values and satisfies two conditions for the optimum approximation presented in this paper. The optimum approximation presented in this paper uses sample values of signal directly. Hence, the presented result is independent from the so-called initial problem in wavelet theory.