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Some properties of zeros of Sobolev-type orthogonal polynomials. (English) Zbl 0862.33005
The authors consider the monic polynomials \(Q_n (n=1,2, \dots)\) which are orthogonal with respect to a certain inner product involving also a discrete part (Sobolev-type product) of the form (1) \(\langle f,g \rangle = \int_Ifgd \mu+ \sum^r_{i=0} M_if^{(i)} (c)g^{(i)} (c)\), where \(\mu\) stands for finite positive Borel measure supported on an interval \(I \subset\mathbb{R}\) while \(c\notin {\overset\circ I}\) (the interior of \(I)\), \(r\geq 1\), \(M_i\geq 0\) \((i=0, \dots, r-1)\) and \(M_r>0\), and derive some properties of their zeros. The results are given in the form of two main theorems. One of them states that \(Q_n\) has at least \((n- \overline n)\) changes of sign in the interior of the convex hull of \(I\), where \(\overline n\) denotes the number of terms in the discrete part in (1) whose order of derivatives is less than \(\overline n\). The other main result concerns the interlacing properties of the zeros and gives the conditions under which \(Q_n\) and \(Q_{n+1}\) have common zeros.

MSC:
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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[1] Alfaro, M.; Marcellán, F.; Meijer, H.G.; Rezola, M.L., Symmetric orthogonal polynomials for Sobolev-type inner products, J. math. anal. appl., 184, 360-381, (1994) · Zbl 0809.42012
[2] Alfaro, M.; Marcellán, F.; Rezola, M.L.; Ronveaux, A., On orthogonal polynomials of Sobolev type: algebraic properties and zeros, SIAM J. math. anal., 23, 737-757, (1992) · Zbl 0764.33003
[3] de Bruin, M.G., A tool for locating zeros of orthogonal polynomials in Sobolev inner product spaces, J. comput. appl. math., 49, 27-35, (1993) · Zbl 0792.42010
[4] Chihara, T.S., An introduction to orthogonal polynomials, (1978), Gordon and Breach New York · Zbl 0389.33008
[5] Koekoek, R., Generalizations of the classical Laguerre polynomials and some q-analogues, () · Zbl 0737.33003
[6] Koekoek, R.; Meijer, H.G., A generalization of Laguerre polynomials, SIAM J. math. anal., 24, 768-773, (1993) · Zbl 0780.33007
[7] López, G., On the convergence of the Padé approximants for meromorphic functions of Stieltjes type, Math. USSR sb., 39, 281-288, (1981) · Zbl 0463.30030
[8] Marcellán, F.; Alfaro, M.; Rezola, M.L., Orthogonal polynomials on Sobolev spaces: old and new directions, J. comput. appl. math., 48, 113-131, (1993) · Zbl 0790.42015
[9] Marcellán, F.; López, G.; Van Assche, W., Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product, Constr. approx., 11, 107-137, (1995) · Zbl 0840.42017
[10] Marcellán, F.; Pérez, T.E.; Piñar, M., On zeros of Sobolev type orthogonal polynomials, Rend. mat. appl., 12, 7, 455-473, (1992) · Zbl 0768.33008
[11] Meijer, H.G., Laguerre polynomials generalized to a certain discrete Sobolev inner product space, J. approx. theory, 73, 1-16, (1993) · Zbl 0771.42015
[12] Peherstorfer, F., Linear combinations of orthogonal polynomials generating positive quadrature formulas, Math. comp., 55, 231-241, (1990) · Zbl 0708.65022
[13] Pérez, T.E.; Piñar, M., Global properties of the zeros for Sobolev-type orthogonal polynomials, J. comput. appl. math., 49, 225-232, (1993) · Zbl 0792.42012
[14] Piñar, M., Polinomios ortogonales tipo Sobolev: aplicaciones, ()
[15] Shohat, J., On mechanical quadratures, in particular, with positive coefficients, Trans. amer. math. soc., 42, 461-496, (1937) · Zbl 0018.11902
[16] Xu, Y., A characterization of positive quadrature formulae, Math. comp., 62, 703-718, (1994) · Zbl 0799.65020
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