
Accession Number : AD0630595
Title : RATIONAL APPROXIMATION OF ARBITRARY REAL FUNCTIONS WITH SPECIFIED WEIGHTS.
Descriptive Note : Technical note (Doctoral thesis),
Corporate Author : ILLINOIS UNIV URBANA ELECTRICAL ENGINEERING RESEARCH LAB
Personal Author(s) : Tang,TaoNan
Report Date : 01 MAY 1960
Pagination or Media Count : 99
Abstract : The problem of obtaining the Tchebysheff approximation of a real continuous function in a closed interval by a polynomial or a rational function under a specified weighting function is treated. Solutions of such problems are obtained by numerical methods involving iterative procedures which may be carried out by modern computing machines. The effect of shifting a zero or several zeros of an error function on the weighted error function itself is obtained by multiplying the amount of shift by the sensitivity, defined as the partial derivative of the weighted error function with respect to the zero shifted. Various techniques are used to equalize (and hence minimize) the extrema of the weighted error. The knowledge of the zero shifting effect on the weighted error is used to determine the amount of shifts in different cases. The successive equalization of the weighted error function at the points of extrema gives an iterative procedure with assured convergence of the process. In the case of polynomial approximation, this method yields a set of linear simultaneous equations to be solved in each cycle. In the case of rational function approximation, it results in a set of nonlinear simultaneous equations which can be solved by certain special techniques. Special cases such as the approximation with equalripple relative error and an approximating polynomial with specified cutoff slope are investigated.
Descriptors : (*APPROXIMATION(MATHEMATICS), *FUNCTIONS(MATHEMATICS)), POLYNOMIALS, FUNCTIONAL ANALYSIS, ITERATIONS, ELECTRICAL NETWORKS
Subject Categories : Theoretical Mathematics
Distribution Statement : APPROVED FOR PUBLIC RELEASE