RATIONAL APPROXIMATION OF ARBITRARY REAL FUNCTIONS WITH SPECIFIED WEIGHTS.
Technical note (Doctoral thesis),
ILLINOIS UNIV URBANA ELECTRICAL ENGINEERING RESEARCH LAB
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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 non-linear simultaneous equations which can be solved by certain special techniques. Special cases such as the approximation with equal-ripple relative error and an approximating polynomial with specified cutoff slope are investigated.
- Theoretical Mathematics