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On best rational approximation of analytic functions. (English) Zbl 1073.30029

The main result contained in this paper is described as follows. Let E be a compact set in the extended complex plane C ¯, n be a nonnegative integer and f:EC be continuous. Let us consider the best rational approximation ρ n (f,E) of f in the uniform metric ·-· E on E by the set R n of rational functions of order at most n, that is, ρ n (f,E)=inf rR n f-r E . Denote by Ω the interior of E and assume that KC ¯ is a compact subset with KΩ. If E has nonempty connected complement, Ω, C ¯K is connected and f is analytic in Ω and nonrational, then

lim sup n k=0 n ρ k (f,K)/ k=0 n ρ k (f,E) 1/n 2 exp(-1/C(E,K));

in particular, lim inf n (ρ n (f,K)/ρ n (f,E)) 1/n exp(-2/C(E,K)). Here C(E,K) denotes the condenser capacity associated with the condenser (E,K). The main result is derived by using the Hankel operator A f,G(ε) :φE 2 (G(ε))P(φf)L 2 (G(ε))E 2 (G(ε)) associated to f and to an adequate family of domains {G(ε):0<ε<1}, where, for every domain G of C, E 2 (G) denotes its Smirnov class and P is the orthogonal projection from L 2 (G) onto E 2 (G).

MSC:
30E10Approximation in the complex domain
41A20Approximation by rational functions
41A50Best approximation, Chebyshev systems