Korzh S.A., Ov\^carenko I.E., Ugrinovsky R.A.
Chebyshev's recursion --- some analytical, computational
and applied aspects
(72K, TeX, plain)
ABSTRACT. We consider polynomials orthogonal with respect to some scalar
product on spaces of polynomials on the real line and unit circle.
A basic problem in the constructive theory of such polynomials is
to determine their three-term recurrence relations. Depending on what is known
about the measure corresponding to a given scalar product, there are different
ways to proceed. If, as is typical in applications, one knows the measure only
through its moment information, the appropriate procedure is an algorithm that
goes back to Chebyshev. The algorithm in effect implements the nonlinear map
from the given moments or modified moments to the desired recurrence
coefficients. We study some algebraic and computational structures
connected with scalar products on spaces of polynomials on the real line and
unit circle and Chebyshev's algorithm. To illustrate up-to-date versions
of the Chebyshev algorithm various applied problems are considered.