Theory of Extended Interpolation Approximation and Upper and Lower Bounds of Error.

Assume that signal f(t) is impressed on N parallel time-invariant liner networks H_m(ω) (m = 1∼N), and consider the uniformly sampled values gm(nT) (m=1∼N; n=0, ± 1, ±2; T > 0) of the output signal gm(t) (m=1∼N). This paper discusses the following problem from a unified viewpoint. The response g(t), when the signal f(t) is passed through the given filter H(ω), is to be approximated by and expression y (t) which is the sum of the forementioned sample value multiplied by time functions ø _mn(t) (m-1 ∼ N; n=0, ± 1, ± 2,). It is assumed that the signal f(t) belongs to the set 1 of the signals for which the weighted square integral of the Fourier spectrum F(ω) is not greater than a positive number A. In the foregoing, ømn(t) is called the interpolation function. First, it is shown that given H_m(ω)(m=1∼N), the time-limited interpolation function, which minimizes the upper limit e_max(t) of the error e(t) =g(t) - y(t) over all f(t) belonging to T, is obtained by shifting the impulse responses ø_m(t) (m=1∼N) of certain N linear time-invariant interpolation filters ø_m (ω) (m=1∼N) and ø_m(ω) (m=1∼N) are optimized so that the measure e_max(t) of the error is minimized, the upper and the lower bounds of the optimal e_max(t) are shown. Considering an application where N is large, the optimization is discussed in the sense that the equivalent multiplicity is reduced at the sacrifice of the approximation error. [ABSTRACT FROM AUTHOR]