2836-7863ISSN (Online)
Volume 4 , Issue 2 , PP: 01-08, 2024 | Cite this article as | XML | Html | PDF | Full Length Article
Agnes Osagie 1 *
Doi: https://doi.org/10.54216/NIF.040201
In this research, we study the problem of determining the degree of approximation of functions using the Hausdorff method, and we can do this by proving the following results:
If f∈Lip(α,p)with α>1/p and be a continues aimost everywhere and 2m periodic function,Then the degree of approximation of (f ) ̃using hausdorff means of conjugate fourier series, is given by:
〗|(|H ̃_((n+λ) ) (f,a)-(f ) ̃(a)|)|_p=0((n+λ)^(1/p-α) )
If f be a 2m periodic function, continues almost everywhere on [–m,m] andbelonging to the class Z_(α,p ),p≥1 .then the degree of approximation of function f of fouier series using hausdorff means,is given by:
E_((n+λ) ) (f)= inf_((n+λ) ) ‖H_((n+λ) )-f‖_(α,p)=0(1/((n+λ) ) ∫_(1/(n+λ))^m▒〖t^(α-2)/v(t) dt〗) (5)
where〖 t〗^αand v the zygmund moduli of continuity sunch that t^α/v(t) positive and monotonic function.
Degree of approximation , Zygmund class , Hausdorff method , Continuous almost everywhere, Fourier series
[1] Rhoades, BE.2014, The degree of approximation of function, and their conjugates, belonging to sever al general Lipschitz classes by Hausdorff matrix means of the Fourier series and conjugate series of a Fourier series. Tamkang J.Math.45 (4), 389-395.
[2] Lal, S: Shireen. 2013, Best approximation of functions of generali zed Zygmund class by Matrix –Euler summability mean of fourier series Bull .Math . Anal. Appl.5 (4), 1-13.
[3] U.Singh, S. Sonkar .2013 , Trigonometric approximation of signals function belonging to weighted Lip t – class by Hausdorff means, J Appl. Funct. Anal. 8.37-44.
[4] H.K.Nigam, a Sharma. 2012, on approximation of conjugate of functions belonging to different classes by product means, Int .J. pure Appl. Math .76 (2)303-316.
[5] V.N.Mishra, K. Khatri, L.N. Mishr2012, product summability transform of conjugate series of fourier series, Int .J. Math. Sci. 20121-13.
[6] Rhoades, BE, Ozkoklu, K, Albayrak , l.2011, On the degree of approximation of function belonging to a Lipschitz class by Hausdorff means of its fourier series .Appl .Math .Comput .217(16), 6868-6871
[7] H.K.Nigam , A.Sharma 2010, On approximation of conjugate of a function belonging to Lip(t,r) class by product summability means of conjugate series of fourier series, Int .J . Contemp. Math.Sci 2673-2683.
[8] Mricz, F.2010, Enlarged Lipschitz and Zygmund classes of functions and fourier transforms East J.Approx.16 (3), 259-271.
[9] Rhoades, BE .2003, on the degree of approximation of functions belonging to a Lipschitz class by Hausdroff means of its fouries series. Tamkang J.Math .34(3),245-247
[10] Das,G, Nath, A, Ray, BK.2002, An estimate of the rate of convergence of fourie 10, Series in the generalized Holder metric.In :Anal. Appl., pp.43-60.
[11] A. Zygmun.2002, Trigonometric series, thirded , Cambridge University press, London.
[12] G.Bachman, L. Narici, E. Beckensties.2000, Fourier and Wavelet Analysis, Springer Verlag,New York.
[13] K. Qureshi.1982, On the degree of approximation of functions belonging to the Lip
[14] Qureshi , K.1982, On the degree of approximation of functions belonging to the class of Lip
