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On Bernstein-Szegö orthogonal polynomials on several intervals. (English) Zbl 0688.42020

Summary: Let \(\ell \in {\mathbb{N}}\), \(a_ 1<a_ 2<...<a_{2\ell}\), \(E_{\ell}=\cup^{\ell}_{k=1}[a_{2k-1},a_{2k}]\), \(H(x)=\prod^{2\ell}_{k=1}(x-a_ k)\) and let \(\rho_{\nu}(x)=c\prod^{\nu^*}_{k=1}(x-w_ k)^{\nu_ k}\) be a real polynomial with \(w_ k\not\in int(E_{\ell})\) for \(k=1,...,\nu^*\) and \(\nu_ k=1\) if \(w_ k\) is a boundary point of \(E_{\ell}\). For given \(\rho_{\nu}\) and \(\epsilon =(\epsilon_ 1,...,\epsilon_{\nu^*})\), \(\epsilon_ k\in \{-1,1\}\), the following linear functional on \({\mathbb{P}}\), \({\mathbb{P}}\) denoting the space of real polynomials, is defined: \[ \Psi_{H,\rho_{\nu},\epsilon}(p)=\int_{E_{\ell}}p(x)\frac{\sqrt{- H(x)}}{\rho_{\nu}(x)}sgn(-\prod^{\ell}_{k=1}(x-a_{2k-1}))dx+ \]
\[ +\sum^{\nu^*}_{k=1}(1-\epsilon_ k)\sum^{\nu_ k}_{j=1}\mu_{j,k}p^{(j-1)}(w_ k), \] where \(\mu_{j,k}'s\) are certain numbers. Polynomials orthogonal with respect to the (not necessarily positive definite) linear functional \(\Psi_{H,\rho_{\nu},\epsilon}\) are characterized. Those polynomials are given the name Bernstein-Szegö orthogonal polynomials on several intervals. Special attention is given to the most interesting case \(\epsilon =(1,...,1)\).

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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