Alexander Teplyaev
A note on the theorems of
M. G. Krein and L. A. Sakhnovich
on
continuous analogs of orthogonal polynomials on the circle.
(256K, .ps)
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 $p>2$.