Rational Approximation

Recently several authors have investigated the question of approximating certain functions by reciprocals of polynomials under the uniform norm on the positive real axis . Perhaps these results have some applications in industry and elsewhere (cf. [1, 36]) . Our present motivation is to give a detailed list of all the known results with simplified proofs in some cases and many new results and finally many open problems . Gonchar's article [14] may be of great help to people interested in finite intervals . Long ago Chebyshev has shown "for any function f(x) continuous on the whole real axis and having the finite limit lim es (x) = C, there exists a sequence of continuous rational functions of the form R,,(x) _ Pn(x)/Q,,(x) (where P.(x) and Q,,,(x) are polynomials of degree n) such that lim II f (x) Rn(x)11,,.(_~,~) , 0." But Chebyshev never discussed the rate of convergence of the error to zero . This kind of result has been obtained by Freud and Szabodas [13] in 1968 . In 1955, Hastings has shown [15] by computation functions such as