 02118 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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