Abstract
In the past few decades, a number of researchers have studied the error estimation of functions in various function spaces and obtained some useful results by using various summability techniques due to their wide applicability in science and engineering. The present study aims to establish a result on degree of approximation of conjugate Fourier series of functions in the generalized Zygmund class by using Euler–Nörlund product mean which generalizes several known results.
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References
A.A. Das, S.K. Paikray, T. Pradhan, Approximation of signals in the weighted Zygmund class via Euler-Hausdorff product summability mean of Fourier series. J. Indian Math. Soc. 87, 22–36 (2020)
G.H. Hardy, Divergent Series, 1st edn. (Oxford University Press, 1949)
B.B. Jena, L.N. Mishra, S.K. Paikray, U.K. Misra, Approximation of signals by general matrix summability with effects of Gibbs phenomenon. Bol. Soc. Parana. Mat. 38, 141–158 (2020)
S. Lal, Shireen, Best approximation of functions of generalized Zygmund class by matrix-Euler summability mean of Fourier series. Bull. Math. Anal. Appl. 5, 1–13 (2013)
S. Lal, A. Mishra, Euler-Hausdorff matrix summability operator and trigonometric approximation of the conjugate of a function belonging to generalized Lipschitz class. J. Inequal. Appl. (2013). Article ID: 59
L. Leindler, Strong approximation and generalized Zygmund class. Acta. Sci. Math. 43, 301–309 (1981)
V.N. Mishra, L.N. Mishra, Trigonometric approximation of signals (functions) in \(L^p,(p \ge 1)\)-norm. Int. J. Contemp. Math. Sci. 7, 909–918 (2012)
V.N. Mishra, K. Khatri, L.N. Mishra, Approximation of functions belonging to \(Lip \Big ( \xi (t),r \Big )\) class by \((N,p_n)(E,q)\)-summability of conjugate series of Fourier series. J. Inequal. Appl. (2012). Article ID-296
V.N. Mishra, K. Khatri, L.N. Mishra, Product \((N,p_n)(C,1)\)-summability of a sequence of a sequence of Fourier coefficients. Math. Sci. (2012). Article ID: 38
L.N. Mishra, V.N. Mishra, K. Khatri, Deepmala, On the trigonometric approximation of signals belonging to generalized weighted Lipschitz class \(W \Big ( L^r, \xi (t)\Big ), (r \ge 1)\)-class by matrix \((C^1, N_p)\) operator of conjugate series of its Fourier series. Appl. Math. Comput. 237, 252–263 (2014)
V.N. Mishra, K. Khatri, L.N. Mishra, Deepmala, Trigonometric approximation of periodic signals belonging to generalized weighted Lipschitz \(W \Big ( L_r, \xi (t)\Big ), (r \ge 1)\)-class by Norlund Euler \((N,p_n)(E,q)\) operator of conjugate series of its Fourier series. J. Class. Anal. 5, 91–105 (2014)
F. Moricz, Enlarged Lipschitz and Zygmund classes of functions and Fourier transforms. East. J. Approx. 16, 259–271 (2010)
F. Moricz, J. Nemeth, Generalized Zygmund classes of functions and strong approximation of Fourier series. Acta Sci. Math. 73, 637–647 (2007)
H.K. Nigam, On approximation in generalized Zygmund class. Demonstr. Math. 52, 370–387 (2019)
H.K. Nigam, K. Sharma, On \((E,1)(C,1)\) summability of Fourier series and its conjugate series. Int. J. Pure. Appl. Math. 82, 365–375 (2013)
B.P. Padhy, P.K. Das, M. Misra, P. Samanta, U.K. Misra, Trigonometric Fourier approximation of the conjugate series of a function of generalized Lipschitz class by product summability, in Advances in Intelligent Systems and Computing, vol. 410 (2016). https://doi.org/10.1007/978-81-322-2734-2-19
T. Pradhan, S.K. Paikray, U.K. Misra, Approximation of signals belonging to generalized Lipschitz class using \((\overline{N}; p_n; q_n)(E; s)\)-summability mean of Fourier series. Cogent Math. 1250343, 1–9 (2016)
T. Pradhan, B.B. Jena, S.K. Paikray, H. Dutta, U.K. Misra, On approximation of the rate of convergence of Fourier series in the generalized Holder metric by deferred N örlund mean. Afr. Mat. 30, 1119–1131 (2019)
T. Pradhan, S.K. Paikray, A.A. Das, H. Dutta, On approximation of signals in the generalized Zygmund class via \((E; 1)(\overline{N}; p_n)\) summability means of conjugate Fourier series. Proyecciones J. Math. 38, 981–998 (2019)
P. Parida, S.K. Paikray, M. Das, U.K. Misra, Degree of approximation by product \((N; p_n; q_n)(E; q)\) summability Fourier series of a signal belonging to \(Lip(\alpha; r)\)-class. TWMS J. Appl. Eng. Math. 9, 901–908 (2019)
P. Parida, S.K. Paikray, H. Dutta, On approximation of signals in \(Lip(\alpha, r)\)-class using the product \((\overline{N}, p_{n}, q_{n})(E, s)\)- summability means of conjugate Fourier series. Nonlinear Stud. 27, 1–9 (2020)
M.V. Singh, M.L. Mittal, B.E. Rhoades, Approximation of functions in the generalized Zygmund class using Hausdorff means. J. Inequal. Appl. 101 (2017). https://doi.org/10.1186/s13600-017-1361-8
A. Zygmund, Trigonometric Series, vol. 1, 2nd revised edn. (Cambridge University Press, Cambridge, 1939)
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Padhy, B.P., Paikray, S.K., Mishra, A., Misra, U.K. (2021). On Approximation of Functions in the Generalized Zygmund Class Using \((E,r)(N,q_n)\) Mean Associated with Conjugate Fourier Series. In: Paikray, S.K., Dutta, H., Mordeson, J.N. (eds) New Trends in Applied Analysis and Computational Mathematics. Advances in Intelligent Systems and Computing, vol 1356. Springer, Singapore. https://doi.org/10.1007/978-981-16-1402-6_16
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