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)

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 $p>2$.