Abstract
Let E be an arbitrary subset of the unit circle T and let f be a function defined on E. When there exist polynomials P n which are uniformly bounded by a number M > 0 on T and converge (pointwise) to f at each point of E? We give a necessary and sufficient description of such functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Danielyan A.A.: On a polynomial approximation problem. J. Approx. Theory 162, 717–722 (2010)
Danielyan, A.A.: The peak-interpolation theorem of Bishop, Complex analysis and dynamical systems IV. Part 1, Contemp. Math., vol. 553, pp. 27–30. Amer. Math. Soc., Providence (2011)
Kolesnikov S.V.: On a certain theorem of M.V. Keldysh concerning the pointwise convergence of sequences of polynomials. Mat. Sb. 124(166), 568–570 (1984)
Koosis P.: Introduction to H p Spaces. Cambridge University Press, Cambridge (1980)
Privalov, I.I.: Boundary Properties of Analytic Functions, 2nd edn. GITTL, Moscow-Leningrad (1950; in Russian) (German edition: Randeigenschaften Analytischer Funktionen, Deutscher Verlag der Wissenschaften; 1956)
Rudin W.: Real and Complex Analysis. McGraw-Hill Book Co., New York (1987)
Rudin W.: Functional Analysis. McGraw-Hill Book Co., New York (1991)
Yosida K.: Functional Analysis, 4th edn. Springer, New York (1974)
Zalcman L.: Polynomial approximation with bounds. J. Approx. Theory 34, 379–382 (1982)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Danielyan, A.A. Weak-Star Convergence and a Polynomial Approximation Problem. Results. Math. 69, 257–262 (2016). https://doi.org/10.1007/s00025-015-0459-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-015-0459-x