Journal for History of Mathematics (한국수학사학회지)

The Korean Society for History of Mathematics (한국수학사학회)
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A Brief Study on Bhatia's Research of L1-Convergence

바티의 L1-수렴성 연구에 관한 소고

  • Received : 2013.12.17
  • Accepted : 2014.02.05
  • Published : 2014.02.28
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Abstract

The $L^1$-convergence of Fourier series problems through additional assumptions for Fourier coefficients were presented by W. H. Young in 1913. We say that they are the classical results. Using modified trigonometric series is the convenience method to study the $L^1$-convergence of Fourier series problems. they are called the neoclassical results. This study concerns with the $L^1$-convergence of Fourier series. We introduce the classical and neoclassical results of $L^1$-convergence sequentially. In particular, we investigate $L^1$-convergence results focused on the results of Bhatia's studies. In conclusion, we present the research minor lineage of Bhatia's studies and compare the classes of $L^1$-convergence mutually.

Keywords

References

  1. S. S. BHATIA, Integrability and $L^1$-convergenece of trigonometric series with special coefficients, Ph. D Thesis, M. D. University, Rohtak, India, 1993.
  2. S. S. BHATIA and B. RAM, On $L^1$-convergence of certain modifed trigonometric sums, Indian J. Math. 35(2) (1993), 171-176.
  3. S. S. BHATIA and B. RAM, On $L^1$-convergence of modifed complex trigonometric sums, Proc. Indian Acad. Sci. Math. Sci. 105(2) (1995), 193-199. https://doi.org/10.1007/BF02880365
  4. S. S. BHATIA and B. RAM, The extensions of the F. Moricz theorems, Proc. Amer. Math. Soc. 6(124) (1996), 1821-1829.
  5. S. S. BHATIA and B. RAM, On weighted Integrability of double cosine series, J. of Mathematical Analysis and Applications 208 (1997), 510-519. https://doi.org/10.1006/jmaa.1997.5348
  6. Kulwinder KAUR and S. S. BHATIA, Integrability and $L^1$-convergence of Rees-Stanojevic sums with generalized semiconvex coefficients, IJMMS 30:11 (2002), 645-650.
  7. N. HOODA, B. RAM and S. S. BHATIA, On $L^1$-convergence of a modified cosine sum, Soochow J. of mathematics 28(3) (2002), 305-310.
  8. K. KAUR, S. S. BHATIA and B. RAM, $L^1$-convergence of complex double Fourier series, Proc. Indian Acad. Sci. Math. Sci. 113(4) (2003), 355-363. https://doi.org/10.1007/BF02829630
  9. S. S. BHATIA, Kulwinder KAUR and B. RAM, $L^1$-convergence of modified complex triginometrc sums, Lobachevskii Journal of Mathematics 12 (2003), 3-10.
  10. K. KAUR, S. S. BHATIA, and B. RAM, On $L^1$-convergence of certain trigonometric sums, Georgian Math. J. 11(1) (2004), 99-104.
  11. K. KAUR, S. S. BHATIA, and B. RAM, Double trigonometric series with coefficients of bounded variation of higher order, Tamkang J. Math. 35(4) (2004), 267-280.
  12. J. KAUR and S. S. BHATIA, Integrability and $L^1$-convergence of certain cosine sums, Kyungpook Mathematical Journal 47 (2007), 323-328.
  13. J. KAUR and S. S. Bhatia, Convergence of new modified trigonometric sums in the metric space $L^1$, The Journal of Nonlinear Sciences and Applications 1(3) (2008), 179-188.
  14. Jatinderdeep KAUR and S. S. BHATIA, The extension of the theorem of J. W. Garrett, C. S. Rees and C. V. Stanojevic from one dimension to two dimension, Int. Journal of Math. Analysis 3(26) (2009), 1251-1257.
  15. J. KAUR, S. S. BHATIA, Convergence of modified complex trigonometric sums in the metric space L, Lobachevskii J. Math. 31(3) (2010), 290-294. https://doi.org/10.1134/S1995080210030121
  16. J. KAUR and S. S. BHATIA, Integrability and $L^1$-convergence of double cosine trigonometric series, Anal. Theory Appl. 27(1) (2011), 32-39. https://doi.org/10.1007/s10496-011-0032-8
  17. J. KAUR, S. S. BHATIA, A class of $L^1$-convergence of new modified cosine sums, Southeast Asian Bull. Math. 36 (2012), 831-836.
  18. LEE Jung Oh, The life of Fourier, The minor Lineage of his younger scholars and a theorem of Telyakovskii on $L^1$-convergence, The Korean Journal for History of Mathematics 22(1) (2009), 25-40.
  19. LEE Jung Oh, Partial sum of Fourier series, the reinterpret of $L^1$-convergence results using Fourier coefficients and theirs minor lineage, The Korean Journal for History of Mathematics 23(1) (2010), 53-66.
  20. LEE Jung Oh, A brief study on Stanojevic's works on the $L^1$-convergence, The Korean Journal for History of Mathematics 26(2-3) (2013), 163-176. https://doi.org/10.14477/jhm.2013.26.2_3.163

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