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Approximation of Signals in the Weighted Zygmund Class via Euler-Hausdorff Product Summability Mean of Fourier Series


Affiliations
1 Department of Mathematics, Veer Surendra Sai University of Technology, Burla, Odisha 768018, India
2 Department of Mathematics, Gauhati University, Guwahati 781014, India
     

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Approximation of functions of Lipschitz and zygmund classes have been considered by various researchers under different summability means. In the proposed paper, we have studied an estimation of the order of convergence of Fourier series in the weighted Zygmund class W(Zr(ω)) by using Euler-Hausdorff product summability mean and accordingly established some (presumably new) results. Moreover, the results obtained here are the generalization of several known results.

Keywords

Degree of Approximation, Weighted Zygmund Class, Trigonometric Fourier Series, Euler Mean, Hausdorff Mean.
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  • A. Aasma, H. Dutta and P. N. Natarajan, An Introductory Course in Summability Theory, John Wiley & Sons, Inc. Hoboken, NJ, USA, 2017.
  • A. A. Das, B. B. Jena, S. K. Paikray and R. K. Jati, Statistical deferred weighted summability and associated Korovokin-type approximation theorem, Nonlinear Sci. Lett. A, 9(3) (2018), 238–245.
  • H. Dutta and B. E. Rhoades (Eds.), Current Topics in Summability Theory and Applications, Springer, Singapore, 2016.
  • G. H. Hardy, Divergent Series, Oxford University Press, Oxford, 1949.
  • B. B. Jena, L.N. Mishra, S.K. Paikray and U.K. Misra, Approximation of signals by general matrix summability with effects of Gibbs phenomenon, Bol. Soc. Paran. Mat. (2018) Doi: 10.5269/bspm.v38i6.39280.
  • B. B. Jena, S. K. Paikray and U. K. Misra, A Tauberian theorem for double Ces`aro summability method, Int. J. Math. Math. Sci., (2016), 1–4.
  • B. B. Jena, S. K. Paikray and U. K. Misra, Inclusion theorems on general convergence and statistical convergence of (L, 1, 1) - summability using generalized Tauberian conditions, Tamsui Oxf. J. Inf. Math. Sci., 31 (2017), 101–115.
  • B. B. Jena, S. K. Paikray and U. K. Misra, Statistical deferred Ces`aro summability and its applications to approximation theorems, Filomat, 32(6)(2018), 2307–2319.
  • B. B. Jena, Vandana, S. K. Paikray and U. K. Misra, On generalized local property of |A; |k-summability of factored Fourier series, Int. J. Anal. Appl., 16 (2018), 209–221.
  • S. Lal and A. Mishra, Euler-Hausd¨orff matrix summability operator and trigonometric approximation of the conjugate of a function belonging to the generalized Lipschitz class, J. Inequal. Appl., 59 (2013), 1–14.
  • S. Lal and Shireen, Best approximation of functions of generalized Zygmund class by matrix-Euler summability mean of Fourier series, Bull. Math. Anal. Appl.5 (2013), 1–13.
  • L. Leindler, Strong approximation and generalized Zygmund class, Acta Sci. Math. 43 (1981), 301–309.
  • M. Misra, P. Palo, B. P. Padhy, P. Samanta and U. K. Misra, Approximation of Fourier series of a function of Lipchitz class by product means, J. Adv. Math. 9(4) (2014), 2475–2483.
  • U. K. Misra, N. C. Sahoo and S. K. Paikray, Summability of a series by Y − | ¯N , qn|k method, Indian J. Maths. Mathematical Sci.3 (2007), 117–126.
  • U. K. Misra, N. C. Sahoo and S. K. Paikray, A Note on |N, p n , |k -Summability, Indian Academy of Mathematics 30 (2008), 481–487.
  • F. Moricz, Enlarged Lipschitz and Zygmund classes of functions and Fourier transforms, East J. Approx. 16(3) (2010), 259–271.
  • F. Moricz and J. Nemeth, Generalized Zygmund classes of functions and strong approximation by Fourier series, Acta Sci. Math. (Szeged), 73 (2007), 637–647.
  • H. K. Nigam and K. Sharma, On (E,1)(C,1) summability of Fourier series and its conjugate series, Int. J. Pure Appl. Math.,82(3) (2013), 365–375.
  • P. Parida, S. K. Paikray, M. Dash and U. K. Misra, Degree of approximation by product (N, pn, qn)(E, q) summability of Fourier series of a signal belonging to Lip(, r)-class, TWMS J. App. Eng. Math., (Accepted).
  • P. Parida, S. K. Paikray, H. Dutta, B. B. Jena and M. Dash, Tauberian theorems for Ces`aro summability of n-th sequences, Filomat, 32(11) (2018), 3993–4004.
  • S. K. Paikray, Degree of approximation by product ( ¯N ; pn; qn)(E; q) summability of Fourier series of a function belonging to lipschitz class, Asian J. Current Res., 1 (2016), 108–113.
  • S. K. Paikray, R. K. Jati, U. K. Misra and N. C. Sahoo, On degree approximation by product means of conjugate series of Fourier series, Bull. Soc. Math. Services and Standards, 1(2012), 12–20.
  • S. K. Paikray, R. K. Jati, U. K. Misra and N. C. Sahoo, On degree approximation of Fourier series by product means, Gen. Math. Notes, 13(2012), 22–30.
  • S. K. Paikray, U. K. Misra and N. C. Sahoo, Trangular matrix summability of a series, African Jour. Math. Comput. Sci. Res., 4(2011), 164–169.
  • S. K. Paikray, U. K. Misra and N. C. Sahoo, Absolute banach summability of a factored Fourier series, Inter. J. Res. Reviews Appl. Sc. 7(2011), 266–276.
  • S. K. Paikray, U. K. Misra and N. C. Sahoo, Absolute banach summability of a factored conjugate series, Gen. Math. Notes, 9(2012), 19–31.
  • T. Pradhan, S. K. Paikray, B. B. Jena and H. Dutta, Statistical deferred weighted B-summability and its applications to associated approximation theorems, J. Inequal. Appl., (2018), 1–21.
  • T. Pradhan, S. K. Paikray and U. K. Misra, Approximation of signals belonging to generalized Lipschitz class using (N, pn, qn)(E, s)-summability mean of Fourier series, Cogent Mathematics, 3(2016), 1–9.
  • S. Sarangi, S. K. Paikray, M. Dash, M. Misra and U. K. Misra, Degree of approximation of conjugate series of a Fourier Series by Hausdr¨off and N¨orlund product summability, Computational Intelligence in Data Mining, 3(2015), 685–692.
  • S. Sarangi, S. K. Paikray, M. Dash and U. K. Misra, Degree of approximation of Fourier series by Hausd¨orff and N¨orlund product means, J. Comput. Modelling, 3(2013), 145– 152.
  • M. V. Singh, M. L. Mittal and B. E. Rhoades, Approximation of functions in the generalized Zygmund class using Hausdorff means, J. Inequal. Appl., 101 (2017), 1–11.
  • H. M. Srivastava, B. B. Jena, S. K. Paikray and U. K. Misra, A certain class of weighted statistical convergence and associated Korovkin type approximation theorems for trigonometric functions, Math. Methods Appl. Sci., 41(2018), 671–683.
  • H. M. Srivastava, B. B. Jena, S. K. Paikray and U. K. Misra, Generalized equi-statistical convergence of the deferred N¨orlund summability and its applications to associated approximation theorems, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Math. (RACSAM), 112(4) (2018), 1487–1501.
  • H. M. Srivastava, B. B. Jena, S. K. Paikray and U. K. Misra, Deferred weighted Astatistical convergence based upon the (p, q)-Lagrange polynomials and its applications to approximation theorems, J. Appl. Anal., 21(2018), 1–16.
  • E. C. Titechmalch, The Theory of Functions, Oxford University Press, Oxford, 1939.

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  • Approximation of Signals in the Weighted Zygmund Class via Euler-Hausdorff Product Summability Mean of Fourier Series

Abstract Views: 31  |  PDF Views: 1

Authors

A. A. Das
Department of Mathematics, Veer Surendra Sai University of Technology, Burla, Odisha 768018, India
S. K. Paikray
Department of Mathematics, Veer Surendra Sai University of Technology, Burla, Odisha 768018, India
T. Pradhan
Department of Mathematics, Veer Surendra Sai University of Technology, Burla, Odisha 768018, India
H. Dutta
Department of Mathematics, Gauhati University, Guwahati 781014, India

Abstract


Approximation of functions of Lipschitz and zygmund classes have been considered by various researchers under different summability means. In the proposed paper, we have studied an estimation of the order of convergence of Fourier series in the weighted Zygmund class W(Zr(ω)) by using Euler-Hausdorff product summability mean and accordingly established some (presumably new) results. Moreover, the results obtained here are the generalization of several known results.

Keywords


Degree of Approximation, Weighted Zygmund Class, Trigonometric Fourier Series, Euler Mean, Hausdorff Mean.

References





DOI: https://doi.org/10.18311/jims%2F2020%2F22506