Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T18:18:50.543Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

Extract

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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