Abstract
Based on an approach of de Branges and the theory of entire functions, we prove two results pertaining to the Bernstein approximation problem, one concerning analytic perturbations of quasianalytic weights and the other dealing with density of polynomials in spaces with nonsymmetric weights. We improve earlier results of V. P. Gurarii and A. Volberg, giving a more complete answer to a question posed by L. Ehrenpreis and S. N. Mergelyan.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. I. Akhiezer,On the weighted approximation of continuous functions by polynomials on the entire real axis, Uspekhi Mat. Nauk11 (1956), 3–43; English translation in AMS Translations (ser. 2)22 (1962), 95–137.
A. A. Borichev,Analytic quasianalyticity and asymptotically holomorphic functions, St. Peterburg Math. J.4 (1992), 259–272.
A. A. Borichev and M. L. Sodin,The Hamburger moment problem and weighted polynomial approximation on discrete subsets of the real line, in preparation.
L. de Branges,The Bernstein problem, Proc. Amer. Math. Soc.10 (1959), 825–832.
L. de Branges,Hilbert Spaces of Entire Functions, Prentice-Hall, Englewood Cliffs, NJ, 1968.
L. Carleson,On Bernstein’s approximation problem, Proc. Amer. Math. Soc.2 (1951), 953–961.
L. Ehrenpreis,Fourier Analysis in Several Complex Variables, Wiley Interscience, New York, 1970.
A. E. Fryntov,A uniqueness theorem related to the weighted approximation, Math. Physics, Analysis and Geometry (Kharkov)1 (1995), 252–264 (Russian).
W. K. Hayman,Subharmonic Functions, Vol. 2, Academic Press, London, 1989.
I. O. Khachatryan,On weighted uniform approximation of continuous functions by polynomials on two rays, Izv. Akad. Nauk Arm.SSR, Ser. Mat.3 (1968), 301–326 (Russian).
P. Koosis,The Logarithmic Integral, I, Cambridge University Press, Cambridge, 1988.
B. Ya. Levin,Distrbution of Zeros of Entire Functions, Amer. Math. Soc, Providence, RI, 1980.
B. Ya. Levin,Density of systems of functions, quasianalyticity and subharmonic majorants, Zap. Nauchn. Seminarov LOMI170 (1989), 102–156 (Russian); English translation in J. Soviet Math.63 (1993).
N. Levinson,Gap and Density Theorems, Amer. Math. Soc. Colloquium Publications, New York, 1940.
P. Malliavin,Approximation polynomiale pondéré et produits canoniques, C. R. Acad. Sci. Paris247 (1958), 2090–2092.
P. Malliavin,Approximation polynomiale pondéré et produits canoniques, inApproximation Theory and Functional Analysis (J. B. Prolla, ed.), North-Holland, Amsterdam, 1979, pp. 237–262.
S. Mandelbrojt,Séries adhérentes. Régularisation des suites. Applications, Gauthier-Villars, Paris, 1952.
S. N. Mergelyan,Weighted approximation by polynomials, Uspekhi Mat. Nauk11 (1956), 107–152; English translation in AMS Translations (ser. 2)10 (1958), 59–106.
A. A. Vagarshakyan,On approximation by polynomials with non-symmetric weight function, Dokl. Akad. Nauk Arm.SSR72 (1981), 259–262 (Russian).
A. L. Volberg,Weighted completeness of polynomials on the line for a strongly non-symmetric weight, Zap. Nauchn. Seminarov LOMI126 (1983) (Russian); English translation in J. Soviet Math.27 (1984), 2458–2462.
A.L. Volberg,Denseness of the polynomials on a system of rays, Soviet Math. Dokl.29 (1984), 342–347.
R. Yulmukhametov,Approximation of subharmonic functions, Anal. Math.11 (1985), 257–282 (Russian).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sodin, M. Which perturbations of quasianalytic weights preserve quasianalyticity? How to use de Branges’ theorem. J. Anal. Math. 69, 293–309 (1996). https://doi.org/10.1007/BF02787111
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02787111