A Study on Approximation of a Conjugate Function 
using Triple Product Mean

Ramesh Kumar; Ajay Kumar

doi:https://doi.org/10.14445/22315373/IJMTT-V72I1P109

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 1 | Year 2026 | Article Id. IJMTT-V72I1P109 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I1P109

A Study on Approximation of a Conjugate Function using Triple Product Mean

Ramesh Kumar, Ajay Kumar

Received	Revised	Accepted	Published
24 Nov 2025	31 Dec 2025	19 Jan 2026	30 Jan 2026

Citation :

Ramesh Kumar, Ajay Kumar, "A Study on Approximation of a Conjugate Function using Triple Product Mean," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 1, pp. 146-150, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I1P109

Abstract

In this paper, for the first time, we introduce the degree of approximation of 2𝜋-periodic functions belonging to the Lip(𝛼,𝑟) (0<𝛼≤1,𝑟≥1) class by using (𝐸,𝑞)𝑇(𝐶,1) Means of their Conjugate Trigonometric Fourier Series.

Keywords

Conjugate trigonometric Fourier Series, Lip(𝛼,𝑟) class, (𝐸,𝑞)𝑇(𝐶,1) Means.

References

[1] Antoni Zygmund, Trigonometric Series, Cambridge University Press, 3^rd Ed., 2002.
[Google Scholar] [Publisher Link]

[2] B.P. Padhy et al., “On Degree of Approximation of Fourier Series by Product Means ,” International Journal of Mathematics and Computation, vol. 19, no. 2, pp. 34-41, 2013.
[Google Scholar] [Publisher Link]

[3] K. Qureshi, “On the Degree of Approximation of Functions Belonging to the Lipschitz Class by Means of Conjugate Series,” Indian Journal of Pure and Applied Mathematics, vol. 12, no. 9, pp. 1120-1123, 1981. [Google Scholar]

[4] Rupesh Kumar Mishra, and Shambhu Kumar Mishra, “Degree of Approximation of the Conjugate of Functions Belonging to Lip (α,r)-Class by (C,1)(E,q)(E,q) Means of Conjugate Fourier Series,” Communications on Applied Nonlinear Analysis, vol. 32, no. 4s, pp. 118-124, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[5] Rupesh Kumar Mishra, and Shambhu Kumar Mishra, “A Study of Euler-Matrix Triple Product Summability Method of the Fourier Series,” IOSR Journal of Mathematics, vol. 21, no. 3, pp. 49-53, 2025.
[Publisher Link]

[6] M.L. Mittal et al., “Approximation of Functions (Signals) Belonging to Lip(ξ(t), p)-Class by Means of Conjugate Fourier Series Using Linear Operators,” Indian Journal of Mathematics, vol. 47, pp. 217-229, 2005. [Publisher Link]

[7] M.L. Mittal, B.E. Rhoades, and Vishnu Narayan Mishra, “Approximation of Signals (Functions) Belonging to the Weighted W(L_P,ξ(t))-Class by Linear Operators,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, no. 1, pp. 1-12, 2006.
[CrossRef] [Google Scholar] [Publisher Link]

[8] Ramesh Kumar, and Ajay Kumar, “Degree of Approximation of the Conjugate of Functions Belonging to Lip Class by Means of Conjugate Fourier Series,” IOSR Journal of Mathematics, vol. 21, no. 2, pp. 68-75, 2025.
[Publisher Link]

[9] Shyam Lal, and Prem Narain Singh, “Degree of Approximation of Conjugate of Lip (α,p) Function by (C,1) (E,1) means of Conjugate Series of a Fourier Series,” Tamkang Journal of Mathematics, vol. 33, no. 3, pp. 269-274, 2002.
[CrossRef] [Google Scholar] [Pubalisher Link]

[10] Smita Sonker, and Uaday Singh, “Degree of Approximation of the Conjugate of Signals (Functions) belonging to Lip(α,r)-Class by (C, 1)(E,q) Means of Conjugate Trigonometric Fourier Series,” Journal of Inequalities and Applications, vol. 2012, pp. 1-7, 2012.
[CrossRef] [Google Scholar] [Publisher Link]