A weak-star convergence result for orthogonal rational functions
Let \(\lambda\) be a finite positive Borel measure on \(I=[-1,1]\) satisfying \(\lambda'>0\) a.e. and \(B=\{\beta_1,\beta_2,\ldots\}\) be a sequence of poles such that \(B\subset{\mathbb R}\cup\{\infty\}\backslash I\) and \(| \beta_k| \geq \rho>1\) for all \(k.\) The space of rational functions with poles in \(\{\beta_1,\ldots,\beta_n\}\) is defined as \({\mathcal L}_n=\)span\(\{b_0,\ldots,b_n\},\) where \(b_0=1,b_k(x)=\prod_{k=1}^n\frac{x}{1-x/\beta_k}.\) Rational functions \(\{\varphi_0,\ldots,\varphi_n\}\) are the orthonormalization of the basis of \(\mathcal L_n\) with respect to \(\lambda.\) The main result: \[ \frac{1-x/\beta_n}{\sqrt{1-1/\beta_n^2}}\varphi_n^2(x)d\lambda(x)\longrightarrow\frac{1}{\pi}\frac{dx}{\sqrt{1-x^2}} \] in the sense of weak-star convergence of measures. Applications to calculations of quadrature error for formulas based on the orthogonal rational functions are discussed
scientific article; zbMATH DE number 2157686