Generalized coherent pairs on the unit circle and Sobolev orthogonal polynomials

A pair of regular Hermitian linear functionals (U, V) is said to be an (M, N)-coherent pair of order m on the unit circle if their corresponding sequences of monic orthogonal polynomials {phi(n)(z)}(n >= 0) and {psi(n)(z)}(n >= 0) satisfy Sigma(M)(i=0) a(i,n)phi((m))(n+m-i)(Z) = Sigma(N)(i=0)b(j,n)psi(n-j)(z), n >= 0, where M,N,m >= 0, a(i,n) and b(j,n), for 0 <= i <= M, 0 <= j <= N, n >= 0, are complex numbers such that a(M,n) not equal 0, n >= M, b(N,n) not equal 0, n >= N, and a(i,n) = b(i,n) = 0, i > n. When m = 1, (U, V) is called a (M, N)-coherent pair on the unit circle. We focus our attention on the Sobolev inner product < p(z), q(z)>(lambda) = < u, p(z)(q) over bar (1/z)> + lambda < V, p((m)) (z)<(q((m)))over bar>(1/z)>, lambda > 0, m is an element of 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.

Generalized coherent pairs on the unit circle and Sobolev orthogonal polynomials Articles