# Approximation of continuous functions on hexagonal domains

## Main Article Content

## Abstract

Some approximation properties of hexagonal Fourier series are investigated. The order of approximation by Nörlund means of hexagonal Fourier series is estimated in terms of modulus of continuity.

Keywords

Hexagonal Fourier series, modulus of continuity, Norlund means, order of approximation

## Article Details

How to Cite

*J. Numer. Anal. Approx. Theory*,

*47*(1), 42-57. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1128

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## References

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[2] S.N. Bernstein, Sur l’ordre de la meilleure approximation des fonctions continues parles polynômes de degré donné, Mem. Cl. Sci. Acad. Roy. Belg., 4(1912), pp. 1–103.

[3] P.L. Butzer, R.J. Nessel, Fourier analysis and Approximation, Academic Press, New-York-London, 1971.

[4] R.A. DeVore, G.G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin,1993.

[5] B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal., 16(1974), pp. 101–121, Article on the journal website: https://doi.org/10.1016/0022-1236(74)90072-x.

[6] A. Guven, Approximation by (C,1) and Abel-Poisson means of Fourier series on hexagonal domains, Math. Inequal. Appl., 16(2013), pp. 175–191.

[7] A. Guven, Approximation by Nörlund means of hexagonal Fourier series, Anal. Theory Appl., 33(2017), pp. 384–400.

[8] G.H. Hardy, Divergent Series, Clarendon Press, Oxford, 1949.

[9] A.S.B. Holland, B.N. Sahney, J. Tzimbalario, On degree of approximation of a class of functions by means of Fourier series, Acta Sci. Math., 38(1976), pp. 69–72.

[10] A.S.B. Holland, A survey of degree of approximation of continuous functions, SIAM Rev.,23(1981), pp. 344–379, Article on the journal website: https://doi.org/10.1137/1023064.

[11] H. Li, J. Sun, Y. Xu, Discrete Fourier analysis, cubature and interpolation on ahexagon and a triangle, SIAM J. Numer. Anal., 46(2008), pp. 1653–1681, Article on the journal website: https:doi.org/10.1137/060671851.

[12] L. Mcfadden, Absolute Nörlund summability, Duke Math. J., 9(1942), pp. 168–207, Article on the journal website: https://doi.org/10.1215/s0012-7094-42-00913-x.

[13] B.N. Sahney, D.S. Goel, On the degree of approximation of continuous functions, Ranchii Univ. Math. J., 4(1973), pp. 50–53.

[14] S.B. Stechkin, The approximation of periodic functions by Fejér sums, Trudy Mat.Inst. Steklov., 62(1961), pp. 48–60 (in Russian).

[15] J. Sun, Multivariate Fourier series over a class of non tensor-product partition domains, J. Comput. Math., 21(2003), pp. 53–62.

[16] A.F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press, New York, 1963.

[17] Y. Xu, Fourier series and approximation on hexagonal and triangular domains, Constr.Approx., 31(2010), pp. 115–138, Article on the journal website: https://doi.org/10.1007/s00365-008-9034-y.

[18] A. Zygmund, Trigonometric Series, 2nd ed., vol. I, Cambridge Univ. Press, New York,1959.

[2] S.N. Bernstein, Sur l’ordre de la meilleure approximation des fonctions continues parles polynômes de degré donné, Mem. Cl. Sci. Acad. Roy. Belg., 4(1912), pp. 1–103.

[3] P.L. Butzer, R.J. Nessel, Fourier analysis and Approximation, Academic Press, New-York-London, 1971.

[4] R.A. DeVore, G.G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin,1993.

[5] B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal., 16(1974), pp. 101–121, Article on the journal website: https://doi.org/10.1016/0022-1236(74)90072-x.

[6] A. Guven, Approximation by (C,1) and Abel-Poisson means of Fourier series on hexagonal domains, Math. Inequal. Appl., 16(2013), pp. 175–191.

[7] A. Guven, Approximation by Nörlund means of hexagonal Fourier series, Anal. Theory Appl., 33(2017), pp. 384–400.

[8] G.H. Hardy, Divergent Series, Clarendon Press, Oxford, 1949.

[9] A.S.B. Holland, B.N. Sahney, J. Tzimbalario, On degree of approximation of a class of functions by means of Fourier series, Acta Sci. Math., 38(1976), pp. 69–72.

[10] A.S.B. Holland, A survey of degree of approximation of continuous functions, SIAM Rev.,23(1981), pp. 344–379, Article on the journal website: https://doi.org/10.1137/1023064.

[11] H. Li, J. Sun, Y. Xu, Discrete Fourier analysis, cubature and interpolation on ahexagon and a triangle, SIAM J. Numer. Anal., 46(2008), pp. 1653–1681, Article on the journal website: https:doi.org/10.1137/060671851.

[12] L. Mcfadden, Absolute Nörlund summability, Duke Math. J., 9(1942), pp. 168–207, Article on the journal website: https://doi.org/10.1215/s0012-7094-42-00913-x.

[13] B.N. Sahney, D.S. Goel, On the degree of approximation of continuous functions, Ranchii Univ. Math. J., 4(1973), pp. 50–53.

[14] S.B. Stechkin, The approximation of periodic functions by Fejér sums, Trudy Mat.Inst. Steklov., 62(1961), pp. 48–60 (in Russian).

[15] J. Sun, Multivariate Fourier series over a class of non tensor-product partition domains, J. Comput. Math., 21(2003), pp. 53–62.

[16] A.F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press, New York, 1963.

[17] Y. Xu, Fourier series and approximation on hexagonal and triangular domains, Constr.Approx., 31(2010), pp. 115–138, Article on the journal website: https://doi.org/10.1007/s00365-008-9034-y.

[18] A. Zygmund, Trigonometric Series, 2nd ed., vol. I, Cambridge Univ. Press, New York,1959.