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In 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 net- works. 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 import- ant arrangements of sapling 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 powereofap 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. [ABSTRACT FROM AUTHOR]

This paper considers approximation formulas of the n-dimensional wave f(X) expressed by the sum of sampled values of f(X) multiplied by certain n-dimensional waves called interpolation functions. A systematic discussion is made to determine whether or not the approximate function converges to the original wave f(X). In this paper it is assumed that the interpolation functions satisfy the causality in regard to a certain axis in the n-dimensional space Rn. Usually, that axis is considered to be the tine axis. These discussions are important from the theoretical viewpoint, since it is closely related to the problem as to whether or not interpolation functions can be realized as impulse responses of physically realizable time-in- variant linear circuits. The set of sampling points is defined as a set of vertices obtained by sampling periodically all the vertices of the n-dimensional parallel polyhedra periodically arranged in the space Rn. This definition of the set of sampling points Includes most of the important arrangements of the sampling points, such as the hexagon and octagon lattices in the two-dimensional plane. In this paper, it is assumed that the interpolation functions have supports, which are parallel translations of each other. The functional forms of the interpolation functions may be different. The set of f(X) considered in this paper is defined as the set of n-dimensional waves whose Fourier spectra have the weighted power-of-p norm (p > 1) smaller than the specified positive value. It is shown first that, under the condition that the squared-sum of the interpolation functions is bounded, the considered approximate expression does not in general converge to the original wave. It is shown that if the I ore-going condition is relaxed, an approximate expression can be found with the impulse response of a causal time-invariant linear network as the interpolation function, which converges to the original wave. [ABSTRACT FROM AUTHOR]

This paper presents a unified theory for the nonuniform sampling theorem for the low-pass band-limited waveform. In most practical cases, the mean interval of the sampling points is in the so-called oversampling state. One can also assume that there exists a certain period in the sampling point sequence. Those properties of the sampling point sequence are assumed in this paper. Yen has derived a sampling theorem for the case where the sampling-point sequence has a period. In his theorem, however, the interpolation function has a spectrum which is a stepwise discontinuous function, not suited to the realization of the interpolation function by a digital or other similar filters. One of the greatest features of the sampling theorem proposed in this paper is that the interpolation function has a continuous spectrum. The problem has been discussed by one of the authors for the case where the sampling frequency and the band-end frequency of the waveform are in an integer ratio. In this paper, the sampling theorem is derived in a unified and integrated form without imposing such a condition. The variance of the approximation error is also derived for the case where a statistically independent additive error is superposed on the sampled value, and it is shown that the proposed sampling theorem is better in that respect. The discussion is extended to the multidimensional band-limited waveform, and a sampling theorem is shown which has an interpolation function with a continuous spectrum. [ABSTRACT FROM AUTHOR]