Schur's algorithm, orthogonal polynomials, and convergence of Wall's continued fractions in \(L^2(\mathbb{T})\). (Q5931920)

This paper has the character of a monograph on orthogonal polynomials on the unit circle \(\mathbb T\) and their connections to continued fractions.NEWLINENEWLINEIt is a well established fact that the Schur algorithm applied to an analytic selfmap \(f\) of the unit disk \(\mathbb D\) produces a finite or infinite continued fraction which converges to \(f\) in \(\mathbb D\). The coefficients of this continued fraction are simple expressions in the Schur parameters \(\{\gamma_n\}\), and the tails of the continued fraction converge to the Schur functions \(f_n\) of \(f\). The canonical denominators of this continued fraction are orthogonal polynomials with respect to a probability measure \(\sigma\) on the unit circle \(\mathbb T\). We let \(\varphi_n\) denote the normalized versions of these polynomials.NEWLINENEWLINEKrushchev proves a long list of nice results, some old and some new, on connections between properties of \(f\), \(\{f_n\}\) (and their continuous extensions to \(\mathbb T\)), \(\{\gamma_n\}\), \(\sigma\) and \(\{\varphi_n\}\). For example:NEWLINENEWLINE\textbf{Theorem 1} NEWLINE\[NEWLINE \begin{aligned} m(\zeta\in{\mathbb{T}}:\,&| f(\zeta)| =1)=0\\ &\Updownarrow\\ \lim_{n\to\infty}\int_{\mathbb T}| f_n| ^2 &dm=0\\ &\Updownarrow\\ \sigma \text{ is an Erdös measure }&\text{(i.e., } \sigma'>0 \text{ a.e. on }\mathbb T). \end{aligned}NEWLINE\]NEWLINE Here \(m\) is the Lebesgue measure.NEWLINENEWLINE\textbf{Theorem 6} NEWLINE\[NEWLINE \begin{aligned} \sigma \text{ is a Rakhmanov measure }&\text{(i.e., } \lim_{n\to\infty}| \varphi_n| ^2 d\sigma=dm)\\ &\Downarrow\\ \lim_{n\to\infty}\int_{\mathbb T}| f-S_n(0)| ^2 &dm=0 \end{aligned}NEWLINE\]NEWLINE where \(S_n(0)\) are the approximants of the continued fraction. (Note that Theorem 6 treats convergence on \(\mathbb T\). The convergence inside \(\mathbb D\) is obvious and well known.)NEWLINENEWLINE\textbf{Theorem 11,12} NEWLINE\[NEWLINE \begin{aligned} \| f_n\| _{\infty}=\mathcal{O}(n^{-\alpha})&\text{ for }0<\alpha<1\\ &\Downarrow\\ \sigma\text{ is absolutely} &\text{ continuous and }\\ (\sigma')^{-1}\in\Lambda_{\alpha}:=\{f\in C(\mathbb T):&\;| f(e^{ix+it})-f(e^{ix})| \leq C_f | t| ^\alpha,\;x,t\in\mathbb R\}\\ &\Downarrow\\ \| f_n\| _{\infty}=&\mathcal{O}\left(\frac{\log n}{n^\alpha}\right). \end{aligned}NEWLINE\]NEWLINE As a ``continued fractionist'' I also want to point out the following result which is interesting in its own right:NEWLINENEWLINE\textbf{Lemma 4.11.} Let \(\{f^k\}\) be a sequence of analytic selfmaps of \(\mathbb D\) which converges locally uniformly to a function \(f\). Let \(\{\gamma_n^k\}\) and \(\{\gamma_n\}\) be the Schur parameters of \(f^k\) and \(f\), respectively. Then NEWLINE\[NEWLINE \lim_{k\to\infty}\gamma_n^k =\gamma_n\quad\text{ for all } n. NEWLINE\]NEWLINE Khrushchev's tools are the classical continued fraction theory, Geronimus' and Szegö's theories on orthogonal polynomials on \(\mathbb T\) and the more modern theories by Rakhmanov, Maté, Nevai, Totik and many others. Everything is put into his own unifying framework with partly new and simpler proofs. Indeed, the paper is very well-written and quite self-contained. Thus it may also serve as an introduction to some or all of these topics.