Abstract
The aim of this paper is to show that several processes studied in trigonometric interpolation theory can be obtained by φ-sums of discrete Fourier series. We shall investigate the uniform convergence of the sequences of thesepolynomials. We show that the convergence of several processes can be seen immediately from suitable explicit forms of the corresponding polynomials. Error estimates for the approximation can be also obtained by certain general results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. N. Bernstein, Collected Works, Vol. II (Moscow, 1954) (in Russian).
S. N. Bernstein, On trigonometric interpolation by the method of least square (in Russian), Doklady Akademii Nauk SSSR, 4 (1934), 1-8.
P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Birkhäuser Verlag (Basel und Stuttgart, 1971).
P. Erdős and G. Halász, On the arithmetic means of Lagrange interpolation, in: Approximation Theory, Kecskemét (Hungary), Coll. Math. Soc. J. Bolyai, 58, (1990), pp. 263-274.
S. Karlin and W. J. Studden, Tchebycheff Systems: with Applications in Analysis and Statistics, Interscience Publishers, A. Division of John Wiley & Sons (New York-London-Sydney, 1974).
O. Kis, On trigonometric (0,2) interpolation, Acta Math. Acad. Sci. Hungar., 11 (1960), 255-276 (in Russian).
O. Kis, Investigation of an interpolation process. I, Acta Math. Acad. Sci. Hungar., 25 (1974), 363-386 (in Russian).
A. Kósa, An Introduction to Mathematical Analysis, Műszaki Könyvkiadó (Budapest, 1981) (in Hungarian).
J. Marcinkiewicz, On interpolation, I, Studia Math., 6 (1936), 1-17 (in French).
I. P. Natanson, Constructive Function Theory, Akadémiai Kiadó (Budapest, 1952) (in Hungarian).
G. M. Natanson and V. V. Zuk, Trigonometric Fourier Series and Approximation Theory, Izdat. Leningrad Unta (Leningrad, 1983) (in Russian).
F. Schipp and J. Bokor, L ∞ system approximation algorithms generated by ϕ summations, Automatica, 33 (1997), 2019-2024.
A. Sharma and A. K. Varma, Trigonometric interpolation, Duke Math. Journal, 32 (1965), 341-357.
J. Szabados, On an interpolatory analogon of the de la Vallée Poussin means, Studia Sci. Math. Hung., 9 (1974), 187-190.
J. Szabados, On the rate of convergence of a lacunary trigonometric interpolation process, Acta Math. Hungar., 47 (1986), 361-370.
J. Szabados and P. Vértesi, Interpolation of Functions, World Sci. Publ. (Singapore-New Jersey-London-Hong Kong, 1990).
L. Szili and P. Vértesi, On uniform convergence of sequences of certain linear operators, Acta Math. Hungar., 91 (2001), 159-186.
P. Vértesi, On certain linear operators. IV, Acta Math. Acad. Sci. Hungar., 23 (1972), 115-125.
A. Zygmund, Trigonometric Series, Vol. I and Vol. II, Cambridge University Press (Cambridge, 1959).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Szili, L. On the Summability of Trigonometric Interpolation Processes. Acta Mathematica Hungarica 91, 131–158 (2001). https://doi.org/10.1023/A:1010691112583
Issue Date:
DOI: https://doi.org/10.1023/A:1010691112583