On approximation of hexagonal Fourier series in the generalized Hölder metric

Year 2020, , 962 - 973, 02.06.2020

Hatice Aslan Ali Güven

https://doi.org/10.15672/hujms.512908

Abstract

Let $f$ be an $H$-periodic continuous function. The approximation order of the function $f$ by deferred Cesaro means of its hexagonal Fourier series is estimated in uniform and generalized H\"{o}lder metrics.

Keywords

Deferred Cesàro means, generalized Hölder class, hexagonal Fourier series

References

• [1] R.P. Agnew, On deferred Cesàro means, Ann. of Math. 33 (3), 413–421, 1932.
• [2] J. Bustamante and M.A. Jimenez, Trends in Hölder approximation, in: Approxima- tion, Optimization and Mathematical Economics, 81–95, Springer, 2001.
• [3] R.A. DeVore and G.G. Lorentz, Constructive Approximation, Springer-Verlag, 1993.
• [4] B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16, 101–121, 1974.
• [5] A. Guven, Approximation by means of hexagonal Fourier series in Hölder norms, J. Class. Anal. 1, 43–52, 2012.
• [6] A. Guven, Approximation by (C, 1) and Abel-Poisson means of Fourier series on hexagonal domains, Math. Inequal. Appl. 16, 175–191, 2013.
• [7] A. Guven, Approximation properties of hexagonal Fourier series in the generalized Hölder metric, Comput. Methods Funct. Theory, 13, 509–531, 2013.
• [8] A. Guven, On approximation of hexagonal Fourier series, Azerb. J. Math. 8, 52–68, 2018.
• [9] A.S.B. Holland, A survey of degree of approximation of continuous functions, SIAM Rev. 23, 344–379, 1981.
• [10] L. Leindler, Generalizations of Prössdorf’s theorems, Studia Sci. Math. Hungar. 14, 431–439, 1979.
• [11] L. Leindler, A relaxed estimate of the degree of approximation by Fourier series in generalized Hölder metric, Anal. Math. 35, 51–60, 2009.
• [12] H. Li, J. Sun and Y. Xu, Discrete Fourier analysis, cubature and interpolation on a hexagon and a triangle, SIAM J. Numer. Anal. 46, 1653–1681, 2008.
• [13] S. Prössdorf, Zur konvergenz der Fourierreihen hölderstetiger funktionen, Math. Nachr. 69, 7–14, 1975.
• [14] J. Sun, Multivariate Fourier series over a class of non tensor-product partition do- mains, J. Comput. Math. 21, 53–62, 2003.
• [15] A.F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press, 1963.
• [16] Y. Xu, Fourier series and approximation on hexagonal and triangular domains, Con- str. Approx. 31, 115–138, 2010.
• [17] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, 1959.