GENERALIZED COHERENT PAIRS ON THE UNIT CIRCLE AND SOBOLEV ORTHOGONAL POLYNOMIALS

Francisco Marcellan and Natalia C. Pinzon-Cortés

Abstract: A pair of regular Hermitian linear functionals $(\U,\V)$ is said to be an \emph{$(M,N)$-coherent pair of order $m$ on the unit circle} if their corresponding sequences of monic orthogonal polynomials $\{\phi_n(z)\}_{n\geq0}$ and $\{\psi_n(z)\}_{n\geq0}$ satisfy $$ \sum_{i=0}^M a_{i,n} \phi^{(m)}_{n+m-i}(z)=\sum_{j=0}^N b_{j,n} \psi_{n-j}(z), \quad n\geq 0, $$ where $M,N,m\geq0$, $a_{i,n}$ and $b_{j,n}$, for $0\leq i\leq M$, $0\leq j\leq N$, $n\geq0$, are complex numbers such that $a_{M,n}\neq0$, $n\geq M$, $b_{N,n}\neq0$, $n\geq N$, and $a_{i,n}=b_{i,n}=0$, $i>n$. When $m=1$, $(\U,\V)$ is called a \emph{$(M,N)$-coherent pair on the unit circle}. We focus our attention on the Sobolev inner product $$ \bigl\langle p(z),q(z)\bigr\rangle_\lambda=\bigl\langle\U, p(z)\overline{q}(1/z) \bigr\rangle +l\bigl\langle\V, p^{(m)}(z)\overline{q^{(m)}}(1/z)\bigr\rangle, \quad \lambda>0, m\in\Z^+, $$ assuming that $\U$ and $\V$ is an $(M,N)$-coherent pair of order $m$ on the unit circle. We generalize and extend several recent results of the framework of Sobolev orthogonal polynomials and their connections with coherent pairs. Besides, we analyze the cases $(M,N)=(1,1)$ and $(M,N)=(1,0)$ in detail. In particular, we illustrate the situation when $\U$ is the Lebesgue linear functional and $\V$ is the Bernstein–Szego linear functional. Finally, a matrix interpretation of $(M,N)$-coherence is given.

Keywords: coherent pairs, Sobolev inner products, structure relations, Hermitian linear functionals, orthogonal polynomials on the unit circle, Sobolev orthogonal polynomials, Lebesgue linear functional, Bernstein–Szego linear functional, Hessenberg matrices

Classification (MSC2000): 13B05; 42C05

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