Inverse problems, Sobolev–Chebyshev polynomials and asymptotics

Abstract

UDC 517.9

Let $(u,v)$ be a pair of quasidefinite and symmetric linear functionals with $\{P_{n}\}_{n\geq0}$ and $\{Q_{n}\}_{n\geq0}$ as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials $\{R_{n}\}_{n\geq0}$ as follows: $$\frac{P_{n+2}'(x)}{n+2}+b_{n}\frac{P_{n}'(x)}{n}-Q_{n+1}(x)=d_{n}R_{n-1}(x),\quad n\geq1.$$

We give necessary and sufficient conditions for $\{R_{n}\}_{n\geq0}$ to be orthogonal with respect to a quasidefinite linear functional $w.$   In addition, we consider the case where $\{P_{n}\}_{n\geq0}$ and $\{Q_{n}\}_{n\geq0}$ are  monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product\begin{equation*}\langle p,q\rangle _{S}=\int\limits _{-1}^{1}pq(1-x^{2})^{-1/2}dx+\lambda_{1}\int\limits _{-1}^{1}p'q'(1-x^{2})^{1/2}dx+\lambda_{2}\int\limits _{-1}^{1}p''q''d\mu(x),\end{equation*} where $\mu$ is a positive Borel measure associated with $w$ and $\lambda_{1},\lambda_{2}>0,$ $\lambda_{2}$ is a linear polynomial of $\lambda_{1}.$