THEORY OF GENERALIZED INTERPOLATORY APPROXIMATION OF MULTI-DIMENSIONAL SIGNALS

Abstract

In this paper, we present a systematic theory of the optimum subband interpolation of a family of n-dimensional signals which are not necessarily band-limited. We assume that the Fourier spectrums of these signals have weighted L2 norms smaller than a given positive number. The proposed method minimizes the measure of error which is equal to the envelope of the approximation errors with respect to the signals. In the following discussion, we assume initially that the infinite number of interpolation functions with different functional forms are used in the corresponding approximation formula. However, the resultant optimum interpolation functions are expressed as the parallel shifts of the finite number of the n-dimensional functions. It should be noted that the optimum interpolation functions presented in this paper satisfy the generalized discrete orthogonality and, as a result, minimize the wide variety of measures of error at the same time. In the literature,6 a similar discussion is presented. However, it is assumed that the signal is band-limited and the interpolation functions are compulsorily time-limited. Hence, these interpolation functions cannot minimize other measures of error except the proposed one. Interesting reciprocal relation in the approximation, is also discussed. An equivalent expression of the approximation formula in the frequency domain is derived.