Convergence of best rational Tchebycheff approximations
BARRY W. BOEHM · Transactions of the American Mathematical Society · 1965
Introduction.To any real-valued function / continuous on the interval [a, b], there exists, for every pair of non-negative integers n, m a unique best Tchebycheff approximation r*(n, ro) among all rational functions of the form a0 + axx +-h anxn rin, ro; x) = b0 + bxx+ ■■■+ bmx"in which the numerator and denominator polynomials are relatively prime and Z\bi\ -Cf. Achieser [ 1 ].In this paper, we investigate the behavior of the error functional Ein,m,f), in terms of the continuity properties of the function /, for various hypotheses on the growth of the degrees n and m of the numerator and denominator polynomials.§2 generalizes some results of Walsh [2] concerning the conditions under which Ein,m,f) approaches zero with increasing n and ro, while §3 gives conditions under which Ein,m,f) does not approach zero; the results in these sections are shown to have counterparts when other fundamental Markoff systems in C[a,b] replace the system {l,x,x2, ••■ ).A theorem of Jackson [3] on Tchebycheff approximation by polynomials states that mn,0,fíSZZZ!"(,,l),where w is the modulus of continuity of / on [a,b]: