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)
Orthogonal polynomials for exponential weights $x^{2\rho} e^{-2Q(x)}$ on [0,$d$). (English) Zbl 1079.42017
This long paper gives a nearly complete treatment of the properties of polynomials orthogonal with respect to exponential weights on (in)finite intervals (bounds, zeros, Christoffel functions etc.). The weights are $$ w(x)=W_{\rho}^2(x)=x^{2\rho}e^{-2Q(x)},\ x\in I=[0,d), $$ with $0<d\leq\infty,\,\rho>-1/2, Q$ continuous and increasing on $I$ with $\lim_{x\uparrow d}\,Q(x)=\infty$. The main results are on -- bounds for the polynomials (Theorems 1.2--1.5 on pages 204/205), -- restricted range inequalities (Theorems 5.1--5.2 on page 220), -- Christoffel functions (Theorems 6.1--6.2 on pages 230/231), -- the zeros (Theorems 7.1--7.2 on page 236). This is a very nicely written paper, entirely within the setting of so-called `hard analysis’.

MSC:
42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type
WorldCat.org
Full Text: DOI
References:
[1] Bonan, S.; Clark, D. S.: Estimates of the Hermite and the freud polynomials. J. approx. Theory 63, 210-224 (1990) · Zbl 0716.42018
[2] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann -- Hilbert Approach, Courant Lecture Notes, 1999.
[3] Deift, P.; Kriecherbauer, T.; Mclaughlin, K.; Venakides, S.; Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. pure appl. Math. 52, 1491-1552 (1999) · Zbl 1026.42024
[4] Freud, G.; Giroux, A.; Rahman, Q. I.: On approximation by polynomials with weight $exp(-x)$. Canad. J. Math. 30, 358-372 (1978) · Zbl 0409.41002
[5] Garnett, J. B.: Bounded analytic functions. (1981) · Zbl 0469.30024
[6] Kasuga, T.; Sakai, R.: Orthonormal polynomials with generalized freud type weights. J. approx. Theory 121, 13-53 (2003) · Zbl 1034.42021
[7] Kriecherbauer, T.; Mclaughlin, K. T. -R.: Strong asymptotics of polynomials orthogonal with respect to freud weights. Internat. math. Res. notices 6, 299-333 (1999) · Zbl 0944.42014
[8] Levin, E.; Lubinsky, D. S.: Orthogonal polynomials for exponential weights. (2001) · Zbl 0997.42011
[9] Lubinsky, D. S.: An extension of the Erdös-Turán inequality for sums of successive fundamental polynomials. Ann. numer. Math. 2, 305-309 (1995) · Zbl 0824.41002
[10] Mastroianni, G.: Polynomial inequalities, functional spaces and best approximation on the real semiaxis with Laguerre weights. Etna 14, 142-151 (2003) · Zbl 1036.41007
[11] Mhaskar, H. N.: Bounds for certain freud-type orthogonal polynomials. J. approx. Theory 63, 238-254 (1990) · Zbl 0716.42020
[12] Mhaskar, H. N.: Introduction to the theory of weighted polynomial approximation. (1996) · Zbl 0948.41500
[13] Mhaskar, H. N.; Saff, E. B.: Extremal problems for polynomials with exponential weights. Trans. amer. Math. soc. 285, 204-234 (1984) · Zbl 0546.41014
[14] Mhaskar, H. N.; Saff, E. B.: Where does the sup norm of a weighted polynomial live?. Constr. approx. 1, 71-91 (1985) · Zbl 0582.41009
[15] Muckenhoupt, B.: Mean convergence of Hermite and Laguerre series II. Trans. amer. Math. soc. 147, 433-460 (1970) · Zbl 0191.07602
[16] Nevai, P.: Orthogonal polynomials. Memoirs amer. Math. soc. 213 (1979) · Zbl 0405.33009
[17] Nevai, P.: Geza freud, orthogonal polynomials and Christoffel functions: a case study. J. approx. Theory 48, 3-167 (1986) · Zbl 0606.42020
[18] Nevai, P.; Totik, V.: Weighted polynomial inequalities. Constr. approx. 2, 113-127 (1986) · Zbl 0604.41014
[19] Nevai, P.; Vertesi, P.: Mean convergence of Hermite -- Fejér interpolation. J. math. Anal. appl. 105, 26-58 (1985) · Zbl 0567.41002
[20] Rakhmanov, E. A.: On asymptotic properties of polynomials orthogonal on the real axis. Math. USSR. Sbornik 47, 155-193 (1984) · Zbl 0522.42018
[21] Saff, E. B.; Totik, V.: Logarithmic potentials with external fields. (1997) · Zbl 0881.31001
[22] Stahl, H. B.; Totik, V.: General orthogonal polynomials. (1992) · Zbl 0791.33009
[23] G. Szegö, Orthogonal Polynomials, American Mathematical Society, Colloquium Publications, vol. 23, American Mathematical Society, Providence, RI, 1975.
[24] V. Totik, Weighted Approximation with Varying Weights, Springer Lecture Notes in Mathematics, vol. 1300, Springer, Berlin, 1994. · Zbl 0808.41001
[25] Vertesi, P.: On an interpolatory inequality of Erdös and Turán and its application. Period. math. Hungar. 40, 195-203 (2000)