Best Uniform Rational Approximations of Functions by Orthoprojections

Let $C[-1,1]$ be the Banach space of continuous complex functions $f$ on the interval $[-1,1]$ equipped with the standard maximum norm $\|f\|$; let $\omega(\,\cdot\,)=\omega(\,\cdot\,,f$ be the modulus of continuity of $f$; and let $R_n=R_n(f)$ be the best uniform approximation of $f$ by rational functions (r.f.) whose degrees do not exceed $n=1,2,\ldots$. The space $C[-1,1]$ is also regarded as a pre-Hilbert space with respect to the inner product given by $(f,g)=(1/\pi)\int_{-1}^1f(x)g(x)(1-x^2)^{-1/2}\,dx$. Let $z_n=\{z_1,z_2,\ldots,z_n\}$ be a set of points located outside the interval $[-1,1]$. By $F(\,\cdot\,,f,z_n)$ we denote an orthoprojection operator acting from the pre-Hilbert space $C[-1,1]$ onto its $(n+1)$-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set $z_n$. In this paper, we show that if $f$ is not a rational function of degree $\leqslant n$, then we can find a set of points $z_n=z_n(f)$ such that $\|f(\,\cdot\,)-F(\,\cdot\,,f,z_n)\|\leqslant 12R_n\ln\frac3{\omega^{-1}(R_n/3)}$.