Skip to main content
SpringerLink
Log in

Degree of Approximation by the \( \left( {T.\,E^{\,1} } \right) \) Means of Conjugate Series of Fourier Series in the Hölder Metric

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions A: Science Aims and scope Submit manuscript

Abstract

We extend the results of Singh and Mahajan (Int J Math Math Sci 2008:9, 2008) which in turn generalizes the result of Lal and Yadav (Bull Cal Math Soc 93:191–196, 2001). In this paper, we determine the degree of approximation of functions \( \widetilde{f\,} \in H_{\omega } , \) a new Banach space using \( \left( {T.\,E^{\,1} } \right) \) summability means of conjugate series of Fourier series. Also, some corollaries have also been deduced from our main theorem and particular cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alexits G (1961) Convergence problems of orthogonal series. Pergamon Elmsford, New York

    MATH  Google Scholar 

  • Bachman G, Narici L, Beckenstien E (2000) Fourier and wavelet analysis. Springer Verlag, New York

    Book  MATH  Google Scholar 

  • Başar F (2011) Summability theory and its applications. Bentham Science Publishers, İstanbul

    Google Scholar 

  • Braha NL (2010) The asymptotic representation for the best approximation of some classes nonperiodic continuous functions. Int J Pure Appl Math 64(1):1–8

    MathSciNet  MATH  Google Scholar 

  • Bustamante J, Roldan CC (2006) Direct and inverse results in Hölder norms. J Approx Theory 138:112–123

    Article  MathSciNet  MATH  Google Scholar 

  • Chandra P (1982) On the generalized Fejér means in the metric of the Hölder space. Math, Nachr 109:39–45

    Article  MathSciNet  MATH  Google Scholar 

  • Chandra P (1990) Degree of approximation of functions in the Hölder metric by Borel’s means. J Math Anal Appl 149:236–246

    Article  MathSciNet  MATH  Google Scholar 

  • Chen JT, Jeng YS (1996) Dual series representation and its applications to a string subjected to support motions. Adv Eng Softw 27(3):227–238

    Article  Google Scholar 

  • Chen JT, Hong H-K, Yeh CS, Chyuan SW (1996) Integral representations and regularizations for a divergent series solution of a beam subjected to support motions. Earthq Engg Struct Dyn 25(9):909–925

    Article  Google Scholar 

  • Das G, Ghosh T, Ray BK (1995) Degree of approximation of functions in the Hölder metric by (e, c) Means. Proc Indian Acad Sci (Math Sci) 105:315–327

    Article  MathSciNet  MATH  Google Scholar 

  • Das G, Ghosh T, Ray BK (1996) Degree of approximation of functions by their Fourier series in the generalized Hölder metric. Proc Indian Acad Sci (Math Sci) 106(2):139–153

    Article  MathSciNet  MATH  Google Scholar 

  • Gil’ MI (2008) Estimates for entries of matrix valued functions of infinite matrices. Math Phys Anal Geom 11:175–186

    Article  MathSciNet  MATH  Google Scholar 

  • Guven A (2012) Approximation by means of hexagonal Fourier series in Hölder norms. J Class Anal 1(1):43–52

    Article  MathSciNet  MATH  Google Scholar 

  • Khan HH (1974a) On the degree of approximation of a functions belonging to the class Lip(α, p). Indian J Pure Appl Math 5:132–136

    MathSciNet  MATH  Google Scholar 

  • Khan HH (1974) Approximation of classes of functions. Ph.D. Thesis, AMU Aligarh

  • Lal S, Yadav KNS (2001) On degree of approximation of function belonging to the Lipschitz class by (C, 1)(E, 1) means of its Fourier series. Bull Cal Math Soc 93:191–196

    MathSciNet  MATH  Google Scholar 

  • Lenski W, Szal B (2015) Approximation of functions from L p(w)β by general linear operators. Acta Math Hungar. https://doi.org/10.1007/s10474-015-0541-6

    Article  MATH  Google Scholar 

  • Mishra VN, Mishra LN (2012) Trigonometric approximation in L p(p ≥ 1) − spaces. Int J Contemp Math Sci 7:909–918

    MathSciNet  MATH  Google Scholar 

  • Mishra VN, Khatri K, Mishra LN (2012) Product (N, p n)(C, 1) summability of a sequence of Fourier coefficients. Math Sci (Springer open access) 6:38

    MATH  Google Scholar 

  • Mishra LN, Mishra VN, Khatri K (2014a) Deepmala, on the trigonometric approximation of signals belonging to generalized weighted Lipschitz W(L r, ξ(t))(r ≥ 1) − class by matrix (C 1. N p) operator of conjugate series of its Fourier series. Appl Math Comput 237:252–2631

    MathSciNet  MATH  Google Scholar 

  • Mishra VN, Khatri K, Mishra LN (2014b) Deepmala, Trigonometric approximation of periodic Signals belonging to generalized weighted Lipschitz W(L r, ξ(t)), (r ≥ 1) − class by Nörlund-Euler (N, p n)(E, q) operator of conjugate series of its Fourier series. J Class Anal 5(2):91–105

    Article  MathSciNet  MATH  Google Scholar 

  • Mittal ML, Rhoades BE (2000) Degree of approximation of functions in a normed space. J Comp Anal Appl 2(1):1–10

    MathSciNet  MATH  Google Scholar 

  • Mohapatra RN, Chandra P (1983) Degree of approximation of functions in the Hölder metric. Acta Math Hung 41(1–2):67–76

    Article  MATH  Google Scholar 

  • Mursaleen M (2014) Applied summability methods. Springer, New York

    Book  MATH  Google Scholar 

  • Mursaleen M, Ahmad R (2013) Korovkin type approximation theorem through statistical lacunary summability. Iran J Sci Technol 37A2:99–102

    MathSciNet  MATH  Google Scholar 

  • Mursaleen M, Mohiuddine SA (2014) Convergence methods for double sequences and applications. Springer, New York

    Book  MATH  Google Scholar 

  • Prössdorf S (1975) Zur konvergenz der Fourier reihen Hölder steliger Funktionen. Math Nachr 69:7–14

    Article  MathSciNet  Google Scholar 

  • Singh T (1992a) Degree of approximation of functions in a normed space. Publ Math Debrecen 40(3–4):261–267

    MathSciNet  MATH  Google Scholar 

  • Singh T (1992b) Approximation to functions in the Hölder metric. Proc Nat Acad Sci India 62(A):224–233

    Google Scholar 

  • Singh T, Mahajan P (2008) Error bound of periodic signals in the Hölder metric. Int J Math Sci 2008:9

    Article  MATH  Google Scholar 

  • Yang J (2015) Norm convergent partial sums of Taylor series. Bull Korean Math Soc 52(5):1729–1735

    Article  MathSciNet  MATH  Google Scholar 

  • Zygmund A (2002) Trigonometric series, 3rd edn. Cambridge Univ. Press, London

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their deep gratitude to the anonymous learned referee(s) and the editor for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this article. Special thanks are due to Dr. F. Moore Editor in Chief & Editors of Iranian Journal of Science and Technology (Sciences), for their kind cooperation, kindness during communication and for their efforts to send the reports of the manuscript timely. The authors are also grateful to all the editorial board members and reviewers of this prestigious journal, i.e., Iranian Journal of Science and Technology (Sciences). The first author acknowledges the Ministry of Human Resource Development, New Delhi, India for supporting this research article. The second author VNM acknowledges that this project was supported by the Cumulative Professional Development Allowance (CPDA), SVNIT, Surat (Gujarat), India. Authors carried out the proof. Each author contributed equally in the development of the manuscript. All the authors conceived the study and participated in its design and coordination. Authors read and approved the final manuscript. The authors declare that there is no conflict of interests regarding the publication of this paper.

Author information

Authors and Affiliations

  1. Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Dumas Road, Surat, Gujarat, 395 007, India

    Kejal Khatri & Vishnu Narayan Mishra

  2. L. 1627 Awadh Puri Colony Beniganj, Phase-III, Opposite-Industrial Training Institute (I.T.I.), Ayodhya Main Road, Faizabad, Uttar Pradesh, 224 001, India

    Vishnu Narayan Mishra

  3. Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh, 484 887, India

    Vishnu Narayan Mishra

Authors
  1. Kejal Khatri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vishnu Narayan Mishra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vishnu Narayan Mishra.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khatri, K., Mishra, V.N. Degree of Approximation by the \( \left( {T.\,E^{\,1} } \right) \) Means of Conjugate Series of Fourier Series in the Hölder Metric. Iran J Sci Technol Trans Sci 43, 1591–1599 (2019). https://doi.org/10.1007/s40995-017-0272-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40995-017-0272-3

Keywords

Mathematics Subject Classification

Search

Navigation