Theory of the optimum approximation of vector-signals with applications

Recently, it has been required to develop efficient method of solving large-scale set of variable-coefficient linear differential equations in the field of the quantum mechanics in order to analyse the 3D structure of prion-protein. In this paper, we present generalized optimum approximation for a certain set of vector-signals that must be useful in solving these differential equations. The presented approximation is quite flexible in choosing sample points and linear preprocessing. The number of variables for a signal and its generalized spectrum are different, in general. In this analysis, we consider the set of vector-signals such that the generalized spectrums have weighted norms smaller than a given positive number. The presented approximation minimizes various worst-case measure of approximation error at the same time among all the linear and the nonlinear approximations under the same conditions.