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The Cauchy integral and analytic continuation*

Published online by Cambridge University Press:  24 October 2008

James. E. Brennan
James. E. Brennan
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY 40506, U.S.A.
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One of the most important concepts in the theory of approximation by analytic functions is that of analytic continuation. In a typical problem, for example, there is generally a region Ω, a Banach space B of functions analytic in Ω and a subfamily ℱ ⊂ B, each member of which is analytic in some larger open set, and one might be asked to decide whether or not ℱ is dense in B. It often happens, however, that either ℱ is dense or the only functions which can be so approximated have a natural analytic continuation across ∂Ω. A similar phenomenon is also known to occur even for approximation on sets without interior. In this article we shall describe a method for proving such theorems which can be applied in a variety of settings and, in particular, to: (1)  the Bernštein problem for weighted polynomial approximation on the real line; (2)  the completeness problem for weighted polynomial approximation on bounded simply connected regions; (3) the Shapiro overconvergence problem for sequences of rational functions with sparse poles; (4) the Akutowicz-Carleson minimum problem for interpolating functions. Although we shall present no new results, the method of proof, which is based on an argument of the author [6], seems sufficiently versatile to warrant exposition.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 3 , May 1985 , pp. 491 - 498
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Ahiezer, N. I.. On weighted approximation of continuous functions by polynomials on the entire number axis. Uspehi Mat. Nauk, 11; no. 4, (1956), 3–43.Google Scholar
[2]Akutowicz, E. J. and Carleson, L.. The analytic continuation of interpolatory functions. J. Analyse Math. 7 (1959/1960), 223–247.CrossRefGoogle Scholar
[3]N, S.. Bernštein. Le problème de l'approximation des fonctions continues sur tout l'axe réel et l'une des ses applications. Bull. Soc. Math. France 52 (1924), 399–410.Google Scholar
[4]Bernštein, S. N.. Collected Works, vol. 1. Izdat. Akad. Nauk SSSR, Moscow, (1952), article 25 and commentary.Google Scholar
[5]Beurling, A.. Analytic continuation across a linear boundary. Acta Math. 128 (1972), 153–182.CrossRefGoogle Scholar
[6]Brennan, J.. Point evaluations, invariant subspaces and approximation in the mean by polynomials. J. Functional Analysis 34 (1979), 407–420.CrossRefGoogle Scholar
[7]Brennan, J.. Weighted polynomial approximation, quasianalyticity and analytic continuation. J. Reine Angew. Math. (in the Press).Google Scholar
[8]Domar, Y.. On the existence of a largest subharmonic minorant of a given function. Ark. Mat. 3 (1958), 429–440.CrossRefGoogle Scholar
[9]Ferrier, J.-P.. Majoration des potentiels du noyau 1/z̄. Inventiones Math. 21 (1973), 311–317.CrossRefGoogle Scholar
[10]Geetha, P. K.. On Bernštein's approximation problem. J. Math. Anal. Appl. 25 (1969), 450–469.CrossRefGoogle Scholar
[11]Hruščev, S. V.. The Brennan alternative for measures with finite entropy. Izv. Akad. Nauk Armjan, SSR 14 (1979), 189–191.Google Scholar
[12]Keldyš, M. V.. Sur l'approximation en moyenne par polynomes des fonctions d'une variable complexe. Mat. Sb. 58 (1945), 1–20.Google Scholar
[13]Mergeljan, S. N.. On the completeness of systems of analytic functions. Uspehi Mat. Nauk 8 (1953), 3–63. English Translation. Amer. Math. Soc. Translations 19 (1962), 109–166.Google Scholar
[14]Mergeljan, S. N.. General metric criteria for the completeness of systems of polynomials. Dokl. Akad. Nauk SSSR 105 (1955), 901–904.Google Scholar
[15]Mergeljan, S. N.. Weighted approximation by polynomials. Uspehi Mat. Nauk 11 (1956), 107–152. English Translation. Amer. Math. Soc. Translations 10 (1958), 59–106.Google Scholar
[16]Shapiro, H. S.. Overconvergence of sequences of rational functions with sparse poles. Ark. Mat. 7, (1968), 343–349.CrossRefGoogle Scholar
[17]Shapiro, H. S.. Generalized analytic continuation. Symposia on Theoretical Physics and Mathematics, vol. 8 (Plenum Press 1968), 151–163.CrossRefGoogle Scholar
[18]Tumarkin, G. Ts.. Conditions for the convergence of the boundary values of analytic functions and approximation on rectifiable curves. Contemporary Problems in the Theory of Analytic Functions (Nauka, Moscow 1966), 283–295.Google Scholar