Theory on extended form of interpolatory approximation of multidimensional waves

Abstract

This paper presents a comprehensive discussion of the approximation of the n-dimensional wave f(X) using the sampled values of the output wave obtained by exciting a series of time-invariant linear circuits by the wave f(X). It is assumed that the approximate wave h(X) of f(X) is given by the sum of sample values of the output wave multiplied by certain n-dimensional waves. For simplicity, n-dimensional waves to be multiplied with the sample values are called the interpolation functions. The set of sample points treated in this paper is defined as a subset obtained by sampling periodically the vertices of the n-dimensional parallelepipeds placed periodically in the space Rn. Such a set of sampling points includes the most of the typical arrangements of the sampling points, such as the hexagonal and the octagonal lattices on the two-dimensional space. It is assumed that the sample values contain statistically independent errors such as the observation error and/or the quantization error. Moreover, it is assumed that the interpolation functions have the supports which are parallel-translations of each other. First, it is assumed that the functional forms of these interpolation functions may be different. Further, a set of n-dimensional waves is considered where the corresponding spectrum has the weighted p-norms smaller than the prescribed positive constant. The standard deviations of the difference between f(X) and their approximations are considered. As the measure of the approximation error, the upper limit of the standard deviation obtained by varying the original waves over the given set of waves is adopted. In the following sections it is shown that the interpolation functions minimizing the forementioned measure of error can be expressed as the parallel-translations of a finite number of functions. Further, in special cases, the interpolation functions have the discrete orthogonality. Since the measure of error is a convex function of the interpolation functions, it is ensured that the global optimum is obtained easily by using the ordinary numerical optimization. For some special cases, the concrete expression for the optimal interpolation functions are derived. Considering the approximation system in the reverse direction, where the linear circuits first passing the input wave are exchanged with the interpolation filters, it is shown that the new interpolation functions also minimize the same measure of error. As a direct consequence, a successive approximation of the interpolation functions is presented, which is suited to the applications such as the multiplex communication of the images of both directions.