The optimum running-type approximation for time-limited worst-case measures of error based on Fredholm integral equation using Pincherle-Goursat kernel

摘要

We begin with a summary of the optimum fixed-type interpolation approximation minimizing the upper bound of various measures of approximation error, simultaneously. The optimum interpolation functions used in this approximation are different from each other and have to cover the entire interval in the time domain to be approximated. Secondly, by applying the above approximation, we present the optimum running-type interpolation approximation for arbitrary long but time-limited signals. The proposed interpolation functions are time-limited and can be realized by FIR filters. Hence, the approximation system can be realized by time-invariant FIR filter bank. We present one-to-one correspondence between error of approximation in a small interval in the time domain and error of approximation in limited but wide interval in the time domain based on Fredholm integral equation using Pincherle-Goursat kernel. Finally, as a practical application of the optimum fixed-type interpolation approximation, we present a discrete numerical solution of differential equations.