Best uniform rational approximations of functions by orthoprojections (Q1889705)

Let \(X\) be one of the following Banach spaces: \(C[- 1, 1]\), the space of complex continuous functions on the interval \([-1,1]\); \(C(\mathbb T)\), the space of complex continuous functions on the circle \(\mathbb T = \{z :| z| = 1\}\); and \(C_A\), the space of functions analytic in the disk \(\mathbb D = \{z :| z| < 1\}\) and continuous on its closure \(\overline{\mathbb D} =\mathbb D\cup\mathbb T\). The space \(X\) is equipped with the standard maximum norm \(\|\cdot\|=\|\cdot\|_X\). Each of the spaces \(X\) is regarded as a pre-Hilbert space. Moreover, in the spaces \(C(\mathbb T)\) and \(C_A\), the inner product \((f,g)\) of functions \(f\) and \(g\) is determined by \((f, g) = (1/\pi)\int_{\mathbb T} f(z)\overline{g(z)}| dz| \). In the space \(C[-1,1]\), the inner product is determined as \((f, g) = (1/\pi)\int_{-1}^1 f(x)\overline{g(x)}(1- x^2)^{1/2}\, dx\). By \(\mathcal R_n\), \(n = 0,1,\dots\), we denote the set of all rational functions \(r\) whose degrees do not exceed \(n\). Let \({\mathbf z}_n =\{z_1, z_2,\dots, z_n\}\) be a set of points of the extended complex plane \(\overline{\mathbb C}\), located outside the interval \([-1,1]\). By \(\mathcal R({\mathbf z}_n)\) we denote the \((n+1)\)-dimensional linear space of rational functions whose poles (with multiplicity taken into account) can only be points of the set \({\mathbf z}_n\). Let \(\mathcal R({\mathbf z}_n)\subset X\). By \(\mathcal F(\cdot,\cdot,{\mathbf z}_n, X)\) we denote the orthoprojection operator acting from the pre-Hilbert space \(X\) into its \((n +1)\)-dimensional subspace \(\mathcal R({\mathbf z}_n)\). The author studies the order of uniform approximations of function from \(X\) by using their orthoprojection onto \(\mathcal R({\mathbf z}_n)\). In this case, the poles \({\mathbf z}_n\) are chosen depending on \(f\). The main result of this paper is the following statement: Suppose that \(f\in X\setminus \mathcal R_n\), \(n\geq1\), \(\omega(\delta) =\omega(\delta,f)\), and \(R_n =R_n(f,X)= \inf\{\| f-r\|:r\subset\mathcal R_n\cap X\}\). Then there exists a set of points \({\mathbf z}_n ={\mathbf z}_n(f)\) such that \(\| f(\cdot)-\mathcal F(\cdot,f,{\mathbf z}_n,X)\|\leq12\,R_n\,\ln\left( 3 /\omega^{-1}(R_n/3)\right). \) Using this result and the well-known upper bounds for \(R_n(f,X)\) for different function classes, the author derives new results about the order of the best rational approximations of functions with bounded variation on \(\mathbb T\) or \([-1,1]\).