Multi-dimensional filter bank systems achieving the highest performance of running approximation under given limited resources

Running approximation uses interpolation functions with limited supports and interpolates a signal in turn from an initial side to the other end of the signal. When continuous worst-case measures of arbitrary operators of error are given, we present the optimum multi-dimensional running approximation of vector-signals in a certain extended vector-filter bank that minimizes various worst-case measures of error, including a measure of error-rate, simultaneously. This multi-dimensional optimum running (scan-type) approximation uses a finite number of sample values which are updated step-like with movement of the variables-vector x on an n-dimensional variables-vector-space. Further, when a family of communication groups having different integer sampling-intervals larger than one is given, we present a solution of maximizing sum of possible band-widths resources of these communication systems by referring Muirhead's theorem in discrete mathematics for Egyptian fractions. We provide an exact summary of past optimum interpolation approximations of multi-dimensional vector-signals, or more exactly, multi-variables-vector-signals, in extended filter banks on a Hilbert space which are given by Kida. Then, we extend past one-dimensional results in [20] to approximation of multi-dimensional vector-signals under the condition that a multi-dimensional transmission filter bank and an arbitrary continuous worst-case-measure of error are given. This problem has been studied for many years, but one result requires complex extension of band-width of signals and others can prove one-dimensional case only. Hence, it is not solved exactly yet with respect to running approximation of multi-dimensional vector-signals. Based on known theorem of k-chromatic number in graph theory, we define a new concept of colored multi-dimensional multi-wing signals and the corresponding multi-dimensional multi-wing approximations that do not interfere each other in colored variables-vector-spaces. We show that backbone of these multi-dimensional multi-wing approximations becomes the presented running approximation. Throughout these discussions, we present a theory of extended multi-dimensional vector filter banks that achieves the highest performance of running approximation under given limited resources.