Conditions of convergence with respect to extended forms of interpolatory approximation for multidimensional waves

Abstract

AbstractIn this paper we consider a family of output waves obtained by impressing simultaneously an n‐dimensional input wave f(X) to the finite number of time‐invariant linear networks. A problem is considered, where the original wave f(X) is to be approximated using the sampled values of the forementioned output waves. A unified theory is presented with respect to the convergence of the approximate wave to the original wave f(X). the discussion given in this paper may be considered as a basis for the future theory of subband coding and multiplexing of the multidimensional signals, in the sense that we use the sampled values of the multidimensional waves transferred through the linear networks. the approximate wave h(X) for f(x) is a sum of the forementioned sample values of these output waves, being multiplied by certain n‐dimensional waves called interpolation functions. the set of sampling points considered in this paper includes most of the practically important arrangements of sampling points, such as hexagonal and octagonal lattices on the two‐dimensional plane. the set of f(X) in this paper is defined as the set of n‐dimensional waves such that the corresponding Fourier spectrum has the weighted power‐of‐p norm (p > 1) which is less than a given positive value. Under several assumptions, sufficient, necessary, and necessary and sufficient conditions are derived for the wave h(X) always to converge to f(X). These convergence conditions are presented which are imposed on the transfer functions of the time‐invariant linear networks through which f(X) propagates as well as the arrangement of the sampling points and the interpolation functions. They are closely related to the rank of the system of linear equations determined by those parameters.