The optimum estimation of statistical signals based on systematic expression of many types of sample arrays in multidimensional space

Abstract

Extended interpolatory approximation is discussed for some classes of <i>n</i>-dimensional statistical signals. Firstly, we present two sufficient conditions of the optimum approximation. Then, as example of this optimum approximation, we consider approximation of <i>n</i>-dimensional statistical signals expressed by linear combination of the finite set of base signals in a <i>n</i>-dimensional space. We assume that these signals have generalized mutual moment smaller than a given positive number. Related topic was discussed in the previous paper. However, discrete running approximation along the time axis that uses shift-invariant interpolation functions with the finite supports is not treated in the previous paper. In the final part of this paper, we discuss best running approximation of <i>n</i>-dimensional signals expressed by linear combination of the finite set of sinusoidal signals in a <i>n</i>-dimensional space. The presented methods have the minimum measure of approximation error among all the linear and the nonlinear approximations using the same measure of error and generalized sample values.