The optimum approximate reconstruction of a signal from the discrete sample values of the prescribed multiple waves

For a set of signals that each signal is defined by means of a certain spectrum-vector composed of a finite number of extended Fourier transforms of component waves, one of the authors presents an extended optimum approximation but a running approximation is not treated. In this paper, we show the outline of the result given in as a premise of the arguments, firstly. Then, under the conditions that the required time-interval in the approximation is wide but limited and the measures of error are continuous, we present the optimum running approximation for this set of signals by using a certain one-to-one correspondence between the error in the wide time-interval and the error in its small segment. It is shown that the presented running approximation minimizes various worst-case measures of approximation error simultaneously and the corresponding interpolation functions are obtained by solving sets of linear equations having constant coefficient-matrices. Finally, we present an example for a multi-input one-output system having separate pass band to eliminate the prescribed noise band.