zbMATH — the first resource for mathematics

Global properties of zeros for Sobolev-type orthogonal polynomials. (English) Zbl 0792.42012
Summary: In this paper we analyze some properties concerning the zeros of orthogonal polynomials \(Q_ n(x)\), associated to the inner product \[ \langle f,g\rangle= \int_ I f(x)g(x)d\mu(x)+ Mf(c)g(c)+ Nf'(c)g'(c), \] where \(I\) is a (not necessarily bounded) real interval, \(\mu\) is a positive measure on \(I\), \(c\in\mathbb{R}\) and \(M\), \(N\geq 0\). In particular, some properties concerning the localization and separation for the roots of \(Q_ n(x)\) are obtained.

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
Full Text: DOI
[1] Alfaro, M.; Marcellán, F.; Rezola, M.L.; Ronveaux, A., On orthogonal polynomials of Sobolev type: algebraic properties and zeros, SIAM J. math. anal., 23, 3, 737-757, (1992) · Zbl 0764.33003
[2] Bavinck, H.; Meijer, H.G., Orthogonal polynomials with respect to a symmetric inner product involving derivatives, Appl. anal., 33, 102-117, (1989) · Zbl 0648.33007
[3] Bromwich, T.J., Introduction to the theory of infinite series, (1965), MacMillan London
[4] Chihara, T.S., An introduction to orthogonal polynomials, (1978), Gordon and Breach New York · Zbl 0389.33008
[5] Koekoek, R., Generalizations of Laguerre polynomials, J. math. anal. appl., 153, 576-590, (1990) · Zbl 0737.33004
[6] Koekoek, R.; Meijer, H.G., A generalization of Laguerre polynomials, SIAM J. math. anal., 24, 3, 768-782, (1993) · Zbl 0780.33007
[7] Marcellán, F.; Pérez, T.E.; Piñar, M.A., On zeros of Sobolev-type orthogonal polynomials, Rend. mat. appl., 12, 7, 455-473, (1992) · Zbl 0768.33008
[8] Marcellán, F.; Ronveaux, A., On a class of polynomials orthogonal with respect to a discrete Sobolev inner product, Indag. math. (N.S.), 1, 451-464, (1990) · Zbl 0732.42016
[9] Meijer, H.G., Laguerre polynomials generalized to a certain discrete Sobolev inner product space, J. approx. theory, 73, 1-16, (1993) · Zbl 0771.42015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.