02-118 Alexander Teplyaev

A note on the theorems of M. G. Krein and L. A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle. (241K, Postscript) Mar 12, 02

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Abstract. Continuous analogs of orthogonal polynomials on the circle are solutions of a canonical system of differential equations, introduced and studied by M.G.Krein and recently generalized to matrix systems by L.A.Sakhnovich. In particular, $\int_{\mathbb R} \frac{|\log\det\tau'(\lambda)|}{1+\lambda^2} d\lambda < \infty$ if and only if $\int_{0}^\infty |P(r,\lambda)|^2 dr < \infty$ for $Im\lambda > 0$, where $\tau'$ is the density of the absolutely continuous component of the spectral measure, and $P(r,\lambda)$ is the continuous analog of orthogonal polynomials. We point out that Krein's and Sakhnovich's papers contain an inaccuracy, which does not undermine known implications from these results, and prove the corrected statement: the convergence of the integrals above is equivalent not to the existence of the limit $\Pi(\lambda) = \lim_{r\to\infty} P_*(r,\lambda)$ but to the convergence of a subsequence. Here $P_*(r,\lambda)$ is the continuous analog of the adjoint polynomials, and $\Pi(\lambda)$ is analytic for $Im\lambda > 0$. The limit as $r\to\infty$ does not necessarily converges even if $\tau$ is absolutely continuous. Also we show that $\Pi(\lambda)$ is unique if the coefficients are in $L^2$, but in general it can be defined only up to a constant multiple even if the coefficients are in $L^p$ for any $p>2$.

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