DEGREE OF APPROXIMATION BY PRODUCT ¯ N, pn, qn E, q SUMMABILITY OF FOURIER SERIES OF A SIGNAL BELONGING TO Lip α, r -CLASS

Yıl 2019, Cilt: 9 Sayı: 4, 901 - 908, 01.12.2019

P. Parida S. K. Paikray M. Dash U. K. Misra

Öz

Approximation of periodic functions by dierent linear summation methods have been studied by many researchers. Further, for sharpening the estimate of errors out of the approximations several product summability methods were introduced by different investigators. In this paper a new theorem has been established on N; pn; qn E; q - summability of Fourier series of a function belonging to f 2 Lip ; r class that generalizes several known results.

Anahtar Kelimeler

Degree of approximation, Fourier series, Lip, r -class, N, pn, qn E, s - mean, Lebesgue integral.

Kaynakça

• Deepmala, Mishra, L. N., Mishra, V. N., (2014), Trigonometric Approximation of Signals (Functions) belonging to the W (Lr, ξ(t)), (r ≥ 1)− class by (E, q) (q > 0)-means of the conjugate series of its Fourier series, Global Journal of Mathematical Sciences, 2, pp. 61-69.
• Hardy, G. H., (1949), Divergent Series, first ed., Oxford University Press, Oxford.
• Lal, S., (2000), On degree of approximation of conjugate of a function belonging to weighted W (Lp, ξ(t)) class by matrix summability mean of conjugate fourier series, Tamkang Journal of Mathematics, 31, pp. 279-288.
• Mishra, V. N., Khan, H. H., Khatri, K., Mishra, L. N., (2013), Degree of approximation of conjugate of signals (functions) belonging to the generalized weighted Lipschitz W0(Lr, ξ(t)), (r ≥ 1)-class by (C, 1)(E, q)-means of conjugate trigonometric Fourier series, Bulletin of Mathematical Analysis and Applications, 5, pp. 40-53.
• Mishra, V. N., Khatri, K., Mishra, L. N., (2012), 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, Article ID 296 doi: 10.1186/1029-242X-2012-296.
• Mishra, V. N., Khatri, K., Mishra, L. N., (2014), Approximation of Functions belonging to the gener- alized Lipschitz Class by C1.Np-summability method of conjugate series of Fourier series, Matematiˇcki Vesnik, 66, pp. 155-164.
• Mishra, V. N., Mishra, L. N., (2012), Trigonometric approximation of signals (functions) in Lp-norm, International Journal of Contemporary Mathematical Sciences, 7, pp. 909 - 918.
• Mishra, V. N., Khatri, K., Mishra, L. N., Deepmala, (2014), Trigonometric approximation of peri- odic Signals belonging to generalized weighted Lipschitz W0(Lr, ξ(t)), (r ≥ 1)-class by N¨orlund-Euler (N, pn)(E, q) operator of conjugate series of its Fourier series, Journal of Classical Analysis, 5, pp. 91-105, doi:10.7153/jca-05-08.
• Mishra, L. N., Mishra, V. N., Khatri, K., Deepmala, (2014), On the trigonometric approximation of signals belonging to generalized weighted Lipschitz W (Lr, ξ(t))(r ≥ 1)− class by matrix (C1.Np) Operator of conjugate series of its Fourier series, Applied Mathematics and Computation, 237, pp. 252-263.
• Misra, M., Palo, P., Padhy, B. P., Samanta, P., Misra, U. K., (2014), Approximation of Fourier series of a function of Lipschitz class by product means, Journal of Advances in Mathematics, 9, pp. 2475-2484.
• Nigam, H. K., (2013), On approximation of functions by product operators, Surveys in Mathematics and its Applications, 8, pp. 125-136.
• Paikray, S. K., Jati, R. K., Misra, U. K., Sahoo, N. C., (2012), On degree of approximation of Fourier series by product means, General Math. Notes., 13, pp. 22-32.
• Pradhan, T., Paikray, S. K., Misra, U. K., (2016), Approximation of signals belonging to generalized Lipschitz class using ( ¯N , p, qn)(E, s)-summability mean of Fourier series, Cogent Mathematics, 2016, 1250343 pp. 1-9.