On degree of approximation of Fourier series of functions in Besov space using Nörlund mean
DOI:
https://doi.org/10.22199/issn.0717-6279-4092Keywords:
Degree of approximation, Banach space, Besov space, Trigonometric Fourier series, Hölder space, (N,rn)-summability meanAbstract
In the present article, we have established a result on degree of approximation of function in the Besov space by (N; rn)- mean of Trigonometric Fourier series
References
G. Alexits, Convergence problem of orthogonal series. New York, NY: Pergamon Press, 1961.
P.Chandra, ”Degree of approximation of functions in the Hölder metric”, The Journal of the Indian mathematical Society, vol. 53, no.1-4, pp. 99-114, 1988.
G. Das, T. Ghosh and B. K. Ray, ”Degree of approximation of function by their Fourier series in the generalized Hölder metric”, Proceedings of the Indian Academy of Sciences-Mathematical Sciences, vol.106, no. 2, pp. 139-153, 1996.
G. H. Hardy, Divergent series, 1st ed. Oxford: Oxford University Press, 1949.
S. Lal and Shireen, ”Best approximation of functions of generalized Zygmund class by Matrix-Euler summability mean of Fourier series”, Bulletin of Mathematical Analysis and Applications, vol. 5, no.4, pp. 1-13, 2013.
L. Leindler, ”Strong approximation and generalized Zygmund class”, Acta Scientiarum Mathematicarum, vol.43, no.3-4, pp. 301-309, 1981.
R. N. Mahapatra and P. Chandra, ”Degree of approximation of functions in the Hölder metric”, Acta Mathematica Hungarica, vol. 41, no.1-2, pp. 67-76, 1983.
V. N. Mishra, K. Khatri, and L. N. Mishra, “Approximation of functions belonging to Lip (ξ(t), r) class by (N, pn)(E,q)-summability of conjugate series of Fourier series”, Journal of Inequalities and Applications, vol. 2012, Art. ID. 296, 2012, https://doi.org/10.1186/1029-242X-2012-296
V. N. Mishra and L. N. Mishra, ”Trigonometric Approximation of signals (functions) in Lp,(p ≥ 1)-norm”, International Journal of Contemporary Mathematical Sciences, vol. 7, no. 17-20, pp. 909-918, 2012.
V. N. Mishra, K. Khatri and L. N. Mishra, ”Product (N, pn)(C, 1)- summability of a sequence of Fourier coefficients”, Mathematical Sciences, vol. 6, Art. ID. 38, 2012, https://doi.org/10.1186/2251-7456-6-38
V. N. Mishra, K. Khatri, L. N. Mishra and Deepmala, ”Trigonometric approximation of periodic signals belonging to generalized weighted Lipschitz W (Lr, ξ(t)) ,(r ≥ 1)-class by Nörlund Euler (N, pn)(E, q) operator of conjugate series of its Fourier series”, Journal of classical Analysis, vol. 5, no.2, pp. 91-105, 2014, https://doi.org/10.7153/jca-05-08
L. N. Mishra, V. N. Mishra, K. Khatri and Deepmala, ”On the Trigonometric approximation of signals belonging to generalized weighted Lipschitz class W (Lr, ξ(t)), (r ≥ 1)-class by matrix (C1, Np) operator of conjugate series of its Fourier series”, Applied Mathematics and Computation, vol. 237, pp. 252-263, 2014, https://doi.org/10.1016/j.amc.2014.03.085
V. N. Mishra and K. Khatri, ”Degree of approximation of functions f ∈ Hw class by the (Np,,E1) means in the Hölder metric”, International Journal of Mathematics and Mathematical Sciences, vol. 2014, Art. ID. 837408, 2014, https://doi.org/10.1155/2014/837408
F. Moricz, ”Enlarged Lipschitz and Zygmund classes of functions and Fourier transforms”, East Journal of Approximations, vol. 16, no.3, pp. 259-271, 2010.
F. Moricz and J. Nemeth, “Generalized Zygmund classes of functions and strong approximation of Fourier series”, Acta Scientiarum Mathematicarum, vol. 73, pp. 637-647, 2007.
L. Nayak, G. Das and B. K. Ray, ”An estimate of the rate of convergence of Fourier series in the generalized Hölder metric by deferred Cesàro mean”, Journal of Mathematical Analysis and Applications, vol. 420, no. 1, pp. 563-575, 2014, https://doi.org/10.1016/j.jmaa.2014.06.001
H. K. Nigam and M. Hadish, ”Best approximation of functions in generalized Hölder class”, Journal of Inequalities and Applications, vol. 2018, Art. ID. 276, https://doi.org/10.1186/s13660-018-1864-y
H. K. Nigam, “On approximation in generalized Zygmund class”, Demonstratio Mathematica, vol. 52, no. 1. pp. 370-387, 2019, https://doi.org/10.1515/dema-2019-0022
B. P. Padhy, P. K. Das, M. Misra, P. Samanta, and U. K. Misra, “Trigonometric Fourier Approximation of the Conjugate Series of a Function of Generalized Lipchitz Class by Product Summability,” in Computational Intelligence in Data Mining, vol. 1, H. Behera and D. Mohapatra, Eds. New Delhi: Springer, 2016, pp. 181–190, https://doi.org/10.1007/978-81-322-2734-2_19
S. Prösdorff, ”Zur Konvergenz der Fourierreihen hölderstetiger Funktionen”, Mathematische Nachrichten, vol. 69, no. 1, pp. 7-14, 1975.
A. D. Ronald and G. Lorentz, Constructive approximation. Berlin: Springer, 1993.
M. V. Singh, M. L. Mittal and B. E. Rhoades, ”Approximation of functions in the generalized Zygmund class using Hausdroff means”, Journal of Inequalities and Applications, vol. 2017, Art. ID. 101, 2017, https://doi.org/10.1186/s13660-017-1361-8
A. Zygmund, ”Smooth functions”, Duke Mathematics Journal, vol. 12, pp. 47-56, 1945.
A. Zygmund, Trigonometric series: Volumes 1 and 2, 2nd ed. Cambridge: Cambridge University Press, 1993.
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