Abstract
Let μ be a compactly supported positive measure on the real line, with associated Christoffel functions λ n (d μ,⋅). Let g be a measurable function that is bounded above and below on supp[μ] by positive constants. We show that λ n (g d μ,⋅)/λ n (d μ,⋅)→g in measure in {x:μ′(x)>0} and consequently in all L p norms, p<∞. The novelty is that there are no local or global restrictions on μ. The main idea is a new maximal function estimate for the “tail” in Nevai’s operators.
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References
Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Orlando (1988)
Breuer, J., Last, Y., Simon, B.: The Nevai condition. Constr. Approx. (to appear)
Criscuolo, G., Mastroianni, G., Nevai, P.: Some convergence estimates of a linear positive operator. In: Chui, C.K., et al. (eds.) Approximation Theory VI, pp. 153–156. Academic Press, San Diego (1989)
Della Vecchia, B.: Uniform approximation by Nevai operators. J. Approx. Theory 116, 28–48 (2002)
Freud, G.: Orthogonal Polynomials. Pergamon Press/Akademiai Kiado, Budapest (1971)
Garnett, J.B.: Bounded Analytic Functions. Academic Press, Orlando (1981)
Lasser, R., Obermaier, J.: Selective approximate identities for orthogonal polynomial sequences. J. Math. Anal. Appl. 279, 339–349 (2003)
Lopez, G.: Relative asymptotics for polynomials orthogonal on the real axis. Math. USSR. Sb. 65, 505–529 (1990)
Maté, A., Nevai, P., Totik, V.: Extensions of Szegő’s theory of orthogonal polynomials, II. Constr. Approx. 3, 51–72 (1987)
Maté, A., Nevai, P., Totik, V.: Extensions of Szegő’s theory of orthogonal polynomials, III. Constr. Approx. 3, 73–96 (1987)
Maté, A., Nevai, P., Totik, V.: Szegő’s extremum problem on the unit circle. Ann. Math. 134, 433–453 (1991)
Nevai, P.: Orthogonal Polynomials. Memoirs of the AMS no. 213 (1979)
Nevai, P., Freud, Geza: Orthogonal polynomials and Christoffel functions: a case study. J. Approx. Theory 48, 3–167 (1986). dom Matrix ensembles. J. Stat. Phys. 86, 109–147 (1997)
Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995)
Rudin, W.: Real and Complex Analysis. McGraw-Hill, Singapore (1986)
Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Springer, New York (1997)
Simon, B.: Orthogonal Polynomials on the Unit Circle, Parts 1 and 2. American Mathematical Society, Providence (2005)
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Communicated by Vilmos Totik.
Research supported by NSF grant DMS0700427 and US-Israel BSF grant 2004353.
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Lubinsky, D.S. A Maximal Function Approach to Christoffel Functions and Nevai’s Operators. Constr Approx 34, 357–369 (2011). https://doi.org/10.1007/s00365-010-9112-9
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DOI: https://doi.org/10.1007/s00365-010-9112-9