Best approximations by rational functions with a fixed number of poles
Let \({\mathcal R}_ n\) be the set of the rational functions of order \(\leq n\) and let \({\mathcal R}_{n,k}\) (\(\hat {\mathcal R}_{n,k})\) be the set of the rational functions \(r\in {\mathcal R}_ n\), which have at most k (0\(\leq k\leq n)\) different geometrical poles in the complex plane C (respectively, in the extended complex plane \(\hat C)\). For every continuous function f on \(\Delta =[0,1]\) we define \(\rho_{n,k}=\rho_{n,k}(f,\Delta)=\inf_{r\in {\mathcal R}_{n,k}}\| f-r\|,\) where \(\|,\|\) is the sup norm, and similarly is defined \({\hat \rho}{}_{n,k}\) with respect to \(\hat {\mathcal R}_{n,k}\). In the present paper the author improves his results [Mat. Sb., Nov. Ser. 86(128), 314-324 (1971; Zbl 0222.41008), Mat. Anal. Prilozh. 53, 67-85 (1975)] concerning the order of \(\rho_{n,k}\) and \({\hat \rho}{}_{n,k}\). He shows that if f is continuous on [0,1] and it admits a multiple-valued analytic continuation to the annulus \(0<| z| <R\), \(R>1\) \((z=0\) is an isolated point of branches of f) and if \(\tau (z)=\tau_ f(z)=| f^+(z)-f^-(z)|\), \(z\in [-1,0]\), where \(f^+(f^-)\) is the analytic continuation of f to the mentioned annulus along the positive (negative) direction with respect to \(z=0\), then for every fixed \(k\geq 1\) hold the two inequalities: \[ \lim_{\overline{n\to \infty}}(\rho_{n,k}/\tau^ 2[-(1/4n)^{4k+2}])>C_ 1,\quad \lim_{\overline{n\to \infty}}(\rho_{n,k}/\tau^ 2[- (1/4n)^{4k}])>C_ 2, \] where \(C_ 1\) and \(C_ 2\) are positive constants. As consequences of this theorem some other estimations for \(\rho_{n,k}\) are obtained.
scientific article; zbMATH DE number 3882960
scientific article