On Bernstein–Szegö Orthogonal Polynomials on Several Intervals

Let $l \in \mathbb{N}$, $a_1 < a_2 < \cdots < a_{2l} $, $E_l = \bigcup _{k = 1}^l [a_{2k - 1} ,a_{2k} ]$, $H(x) = \prod _{k = 1}^{2l} (x - a_k )$ and let $\rho _\nu (x) = c\prod _{k = 1}^{\nu ^ * } (x - w_k )^{\nu _k } $ be a real polynomial with $w_k \notin \operatorname{int} (E_l )$ for $k = 1, \cdots ,\nu ^ * $ and $\nu _k = 1$ if $w_k $ is a boundary point of $E_l $. For given $\rho _\nu $ and $\varepsilon = (\varepsilon _1 , \cdots ,\varepsilon _{\nu ^ * } )$, $\varepsilon _k \in \{ - 1,1\} $, the following linear functional on $\mathbb{P}$, $\mathbb{P}$ denoting the space of real polynomials, is defined: \[ \begin{gathered} \Psi _{H,\rho _\nu ,\varepsilon } (p) = \int_{E_l } {p(x)} \frac{{\sqrt { - H(x)} }}{{\rho _\nu (x)}}\operatorname{sgn} \left( { - \mathop \prod \limits_{k = 1}^l \left( {x - a_{2k - 1} } \right)} \right)dx \hfill \\ \qquad \qquad \qquad + \sum_{k = 1}^{\nu ^ * } {\left( {1 - \varepsilon _k } \right)} \sum\limits_{j = 1}^{\nu _k } {\mu _{j,k} } p^{(j - 1)} \left( {w_k } \right) \hfill \\ \end{gathered} \] where $\mu _{j,k} $’s are certain numbers. Polynomials orthogonal with respect to the (not necessarily positive definite) linear functional $\Psi _{H,\rho _\nu ,\varepsilon } $ are characterized. Those polynomials are given the name Bernstein–Szegö orthogonal polynomials on several intervals. Special attention is given to the most interesting case $\varepsilon = (1, \cdots ,1)$.