A Simple Proof of the Generalized Optimum Continuous Running-Approximation Based on a Class of Multi-Legged-Type Signals

In this paper, with respect to band-limited signals f(t) contained in a given set of signals Ξ, we present very general optimum continuous running approximations that minimize various continuous worst-case measures of running approximation error simultaneously. Firstly, as a means of proof, we consider a multi-legged-type signal m(t) that is a combined-signal of a given infinite number of one-dimensional band-limited signals h_n(t) (n=...,-2,-1,0,1,2,...) in Ξ. A series of given finite segments σ_n(t) of h_n(t) are arranged sequentially and make backbone of m(t). Sets of two other signals of h_n(t) make feet of m(t). With respect to these h_n(t), we consider a series of the Kida´s optimum approximations, g_n(t), each of which uses a given finite number of generalized sample values of h_n(t). Also, we define a similar multi-legged-type approximation y(t) of m(t) using these g_n(t). Secondly, under a slight modification of Ξ, when the backbone of m(t) itself is a band-limited signal f(t) in Ξ, we prove that backbone of y(t) becomes the corresponding optimum continuous running approximation of f(t). Although this paper treats pure theoretical topics, we believe that it is important for multi-paths communication and signal processing systems, such as MIMO systems, to present a fundamental method of constructing the optimum running approximation including various extended multi-paths transmission systems without using difficult high-level mathematical concept.