In this paper, with respect to band-limited signals $f(t)$ contained in a given set of signals $\Xi$, 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 $h_n(t)$~$(n=\ldots,-2,-1,0,1,2,\ldots)$ in $\Xi$. A series of given finite segments $\sigma_n(t)$ 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 $\Xi$, when the backbone of $m(t)$ itself is a band-limited signal $f(t)$ in $\Xi$, 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.
['Yuichi Kida', 'Takuro Kida']
A Simple Proof of the Generalized Optimum Continuous Running-Approximation Based on a Class of Multi-Legged-Type Signals