IJPAM: Volume 120, No. 3 (2018)

Title

ON EQUIVALENCE OF MODIFIED TRIGONOMETRIC SUMS

Authors

Karanvir Singh$^1$, Kanak Modi$^2$
$^1$Department of Applied Mathematics
GZS Campus College of Engineering and Technology
Maharaja Ranjit Singh Punjab Technical University
Bathinda, Punjab, INDIA
$^2$Department of Mathematics
Amity University of Rajasthan
Jaipur, INDIA

Abstract

We establish $L^{1}-$convergence equivalence of modified sums introduced by Rees and Stanojević and Kumari and Ram. It is shown that all the results regarding integrability and $L^{1}$-convergence of cosine series (or sine series) which have been established by different authors so far by using modified cosine sums or sine sums of Rees and Stanojević can also be proved by considering the corresponding sums introduced by Kumari and Ram under same classes of coefficients. We also introduce modified cosine and sine sums and compare them with modified sums introduced by Kaur and Bhatia.

History

Received: January 28, 2017
Revised: July 17, 2018
Published: November 27, 2018

AMS Classification, Key Words

AMS Subject Classification: 42A20, 42A32
Key Words and Phrases: $L^{1}-$convergence, Fez$\acute{e}$r kernel, modified cosine sums

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.

Bibliography

1
Bary N.K., A treatise on trigonometric series,Vol.I and Vol.II, Pergamon Press, London 1964.

2
Bor H., Convergence of certain cosine sums in the metric space $L^{1}$, J. Sci. Eng., 2(1986), 123-128.

3
Bor H., On $L^{1}$-convergence of a modified cosine sum, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No.3, Dec(1992), 235-238.

4
Bhatia S.S. and Ram B., The extensions of F.Móricz theorems, Proc. Amer. Math.Soc., 6 (124)(1996), 1821-1829.

5
Braha N. L., Integrability and $L^{1}$-convergence of certain cosine sums with third quasi hyper convex coefficients, Hacettepe J. of Mathematics and statistics, Vol. 42(6)(2013), 653-658.

6
Chen C.P., $L^1$-convergence of Fourier series, J. Austral.Math. Soc. (Series A), 41 (1986), 376-390.

7
Fomin G.A., On linear methods for summing Fourier series, Math. Sbornik, 66(1964), 144-152.

8
Fomin G.A., A class of trigonometric series, Math. Zametki 23 (1978), 213-222.

9
Garrett J.W. and Stanojevi$\check{c}$ $\check{C}$.V., On integrabilityand $L^1$-convergence of certain cosine sums, Notices, Amer. Math. Soc., 22 (1975), A-166.

10
Garrett J.W. and Stanojevi$\check{c}$ $\check{C}$.V., Necessary and sufficientcondition for $L^1$-convergence of trigonometric series, Proc. Amer. Math. Soc., 60 (1976), 68-71.

11
Garrett J.W. and Stanojevi$\check{c}$ $\check{C}$.V., On $L^1$-convergenceof certain cosine sums, Proc. American Math. Soc., 54 (1976), 101-105.

12
Kaur K., Bhatia S. S. and Ram B., On $L^{1}$-Convergence of certain Trigonometric Sums,Georgian journal of Mathematics, 1(11)(2004), 98-104.

13
Krasniqi X.Z., On $L^{1}$-convergence of sine and cosine modified sums, Journal of Numerical Mathematics and Stochastics, 7(1) (2015), 94-102.

14
Kaur J. and Bhatia S. S., On $L^{1}$-convergence of certain trigonometric sums, Global Journal of Pure and Applied Mathematics, 2(2) (2006), 111-116.

15
Kaur J. and Bhatia S. S., Integrability and $L^{1}$-convergence of certain cosine sums, Kyungpook Mathematical Journal, 47 (2007), 323-328.

16
Kaur J. and Bhatia S. S., Convergence of new modified trigonometric sums in the metric space L, The Journal of Non Linear Sciences and Applications, 1(3)(2008),179-188.

17
Kolmogorov A. N., Sur l'ordere de grandeur des coefficients dela series de Fourier - Lebesque, Bull. Polon. Sci. Ser. Sci. Math. Astronom.Phys., (1923), 83-86.

18
Kumari S. and Ram B., $L^{1}$-convergence of a modified cosine sum,Indian Journal of Pure and Applied Math., 19 (1988), 1101-1104.

19
Móricz F., On the integrability and $L^1$-convergence of sine series,Studia Mathematica T., XCII (1989), 187-200.

20
Ram B., Convergence of certain cosine sums in the metric space L, Proc. Amer. Math. Soc., 66 (2) (1977), 258-260.

21
Rees C.S. and Stanojevi$\check{c}$ $\check{C}$.V., Necessary and Sufficient condition for the integrability of certain cosine sums, J. Math. Anal. Appl., 43 (1973), 579-586.

22
Singh N. and Sharma K.M., $L^{1}$-convergence of modified cosine sums with generalised quasi-convex coefficients, J. Math. Anal. App., 136 (1988), 189-200.

23
Singh N. and Sharma K.M., Convergence of certain sums in the metric space $L^{1}$ ,Proc. American Math. Soc., 72 (1978), 117-120.

24
Sheng S., The extension of theorems of $\check{C}$.V. Stanojevi$\check{c}$ and V.B Stanojevi$\check{c}$ , Proc. Amer. Math. Soc., 110 (1990), 895-904.

25
Sidon S., Hinreichende Bedingungen f$\ddot{u}$r den Fourier-Charakter einer Trigonometrischen Reihe, J. London Math. Soc., 14 (1939), 158-160.

26
Telyakovskii S.A., On a sufficient condition of Sidon for the integrability of trigonometric series, Math. Zametki ,14 (1973), 317-328.

27
Tikhonov S., On $L_{1}$-convergence of Fourier series, J. Math. Anal. App., 347 (2008), 416-427.

28
Tomovski $\check{Z}$., An extension of the Garret-Stanojevi$\check{c}$ class, Approx. Theory and Appl., 16 (1) (2000), 46-51.

29
Tomovski $\check{Z}$., An extension of the Sidon-Fomin type inequality and its applications, Math. Ineq. and Applications, 4 (2) (2001), 231-238.

30
Young W.H., On the Fourier series of bounded functions, Proc. London Math. Soc., 12 (2) (1913), 41-70.

31
Zahid S. and Hasan S., Integrability of Rees-Stanojevi$\check{c}$ sums, Acta Sci. Math.(Szeged), 49 (1985), 295-298.

32
Zygmund A., Trigonometric series, Vol. I, Vol. II, Univ. Press of Cambridge (1959).

How to Cite?

DOI: 10.12732/ijpam.v120i3.5 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 120
Issue: 3
Pages: 339 - 349


Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).