Optimum discrete approximation of multidimensional band-limited signals

We present a necessary and sufficient condition that a given n-dimensional generalized interpolation approximation minimizes various worst-case measures of error of approximation at the same time among all the approximations, including nonlinear approximation, using the same set of sample values. As a typical example of the optimum approximation satisfying the above necessary and sufficient condition, we present n-dimensional generalitd interpolation approximation using the finite number of sample values. Then, we consider n-dimensional generalized discrete interpolation approximation based on n-dimensional FIR filter banks that uses the finite number of sample values in the approximation of each pixel of image but scan the image over the whole pixels. For this scanning-type discrete approximation, we prove that discrete interpolation functions exist that minimize various measures of error of approximation defined at discrete sample points xp=p, simultaneously, where p are the n-dimensional integer vectors. The presented discrete interpolation functions vanish outside the prescribed domain in the integer-vector space. Hence, these interpolation functions are realized by n-dimensional FIR filters. In this discussion, we prove that there exist continuous interpolation functions with extended band-width that interpolate the above discrete interpolation functions and satisfy the condition called discrete orthogonality. This condition is one of the two conditions that constitute the necessary and sufficient condition presented in this paper. Several discrete approximations are presented that satisfy both the conditions constituting the necessary and sufficient condition presented in this paper. The above discrete interpolation functions have much flexibility in their frequency characteristics if appropriate analysis filters are selected.