# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 42C05 General theory of orthogonal functions and polynomials 33C45 Orthogonal polynomials and functions of hypergeometric type
##### Keywords:
orthogonal polynomials; exponential weights
Full Text:
##### References:
 [1] Bauldry, W. C.: Estimates of Christoffel functions of generalized freud-type-weights. J. approx. Theory 46, 217-229 (1986) · Zbl 0626.42014 [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