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Orthonormal polynomials with generalized Freud-type weights. (English) Zbl 1034.42021
The authors study polynomials, orthonormal with respect to a generalized Freud-type weight $W_{rQ}^2(x)=\vert x\vert ^{2r}\exp(-2Q(x)),$ where $r>-1/2$ and $\exp(-2Q(x))$ is a Freud weight. They prove infinite-finite range inequalities, estimates of Christoffel functions, the largest zeros and spacing of zeros for those orthonormal polynomials. The results extend and improve corresponding results for other classes of weights [see {\it E.Levin} and {\it D. S. Lubinsky}, “Orthogonal polynomials for exponential weights” (2001; Zbl 0997.42011)].

MSC:
42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type
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References:
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[2] Freud, G.: Orthogonal polynomials, akademial kiado. (1971)
[3] Freud, G.: On estimates of the greatest zeros of orthogonal polynomials. Acta math. Acad. sci. Hungary 25, 99-107 (1974) · Zbl 0276.42006
[4] Knopfmacher, A.; Lubinsky, D. S.: Mean convergence of Lagrange interpolation for freud’s weights with application to product integration rules. J. comput. Appl. math. 17, 79-103 (1987) · Zbl 0634.41003
[5] Y. Kanjin, R. Sakai, Pointwise convergence of Hermite--Fejér interpolation of higher order for Freud weights, Tôhoku Math. J. 46 (1994) 181--206. · Zbl 0807.41004
[6] Levin, A. L.; Lubinsky, D. S.: Christoffel functions, orthogonal polynomials, and nevai’s conjecture for freud weights. Constr. approx. 8, 463-535 (1992) · Zbl 0762.41011
[7] Lubinsky, D. S.: Gaussian quadrature, weights on the whole real line, and even entire functions with non-negative even order derivatives. J. approx. Theory 46, 297-313 (1986) · Zbl 0608.41017
[8] Lubinsky, D. S.: Strong asymptotics for extremal errors and polynomials associated with Erdös-type weights. Pitman research notes in mathematics 202 (1989) · Zbl 0743.41001
[9] Nevai, P.; Freud, Géze: Orthogonal polynomials and Christoffel functions. A case study. J. approx. Theory 48, 3-167 (1986) · Zbl 0606.42020
[10] Szegö, G. A.: Orthogonal polynomials. American mathematical society colloquium publications 23 (1975) · Zbl 0305.42011