The Impact of Poles on the Convergence of Best Rational Approximants

摘要

The convergence behavior of best uniform rational approximants r_(n,m)~* with numerator degree n and denominator degree m on a compact set E is investigated for functions f continuous on E and ray sequences above the diagonal of the Walsh table, i. e. sequences {n, m(n)}_( n=1)~x withrnlim inf_(n→∞)n/m(n)≥c>1, rnIf the sequence {r_(n, m(n)~*)}_(n=1)~∞ satisfies a pole condition (P_τ),τ > 1, i. e. for any (n. m(n)) there exists a pole ε_n of r_(n, m(n))~* with G(ε_n, ∞) > lon τ where G(z. ∞) is Green's function of C E with pole at ∞, then only two situations can occur: ( 1 ) For any continuum S is contained in C Ernlim inf_(n→∞)‖r_(n, m(n))~*‖_s=∞rnor lim inf_(n→∞)‖f-r_(n, m(n))~*‖_E~(1)<1.