Journal for History of Mathematics (한국수학사학회지)

The Korean Society for History of Mathematics (한국수학사학회)
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On Classical Studies for Summability and Convergence of Double Fourier Series

이중 푸리에 급수의 총합가능성과 수렴성에 대한 고전적인 연구들에 관하여

  • Received : 2014.03.24
  • Accepted : 2014.07.30
  • Published : 2014.08.31
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Abstract

G. H. Hardy laid the foundation of classical studies on double Fourier series at the beginning of the 20th century. In this paper we are concerned not only with Fourier series but more generally with trigonometric series. We consider Norlund means and Cesaro summation method for double Fourier Series. In section 2, we investigate the classical results on the summability and the convergence of double Fourier series from G. H. Hardy to P. Sjolin in the mid-20th century. This study concerns with the $L^1(T^2)$-convergence of double Fourier series fundamentally. In conclusion, there are the features of the classical results by comparing and reinterpreting the theorems about double Fourier series mutually.

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References

  1. Komaravolu CHANDRASEKHARAN and Subbaramiah Minakshisundaram, Some results on double Fourier series, Duke Math. J. 4(3) (1947), 731-753.
  2. Vladimir G. CHELIDZE, On the absolute convergence of double Fourier series, Doklady AN SSSR. 54(2) (1946), 117-120.
  3. CHOW Yuan Shih, On the Cesaro summability of double Fourier series, Tohoku Math. J. 5(3) (1954), 277-283. https://doi.org/10.2748/tmj/1178245273
  4. John J. GERGEN, Convergence criteria for double Fourier series, Trans. Amer. Math. Soc. 35 (1933), 29-63. https://doi.org/10.1090/S0002-9947-1933-1501671-1
  5. John J. GERGEN, Summability of double Fourier series, Duke Math. J. 3 (1937), 133-148. https://doi.org/10.1215/S0012-7094-37-00310-7
  6. Richard P. GOSSELIN, A convergence theorem for double $L^2$ Fourier series. Can. J. Math 10 (1958), 392-398. https://doi.org/10.4153/CJM-1958-038-1
  7. Albert E. GREEN, Double Fourier series and boundary value problems, Mathematical Proceedings of the Cambridge Philosophical Society 40(3) (1944), 222-228. https://doi.org/10.1017/S0305004100018375
  8. Godfrey H. HARDY, On double Fourier series and especially those which represent the double zeta-function with real and incommensurable parameters, Quart. J. Math. 37(1) (1906), 53-79.
  9. John G. HERRIOT, Norlund Summability of Double Fourier Series, Trans. Amer. Math. Soc. 52(1) (1942), 72-94.
  10. 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.
  11. 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.
  12. LEE Jung Oh, A brief study on Stanojevic's works on the $L^1$-convergence, Journal for History of Mathematics 26(2-3) (2013), 163-176. https://doi.org/10.14477/jhm.2013.26.2_3.163
  13. LEE Jung Oh, A brief study on Bhatia's research of $L^1$-convergence, Journal for History of Mathematics 27(1) (2014), 1-13. https://doi.org/10.14477/jhm.2014.27.1.001
  14. Jozef MARCINKIEWICZ, Antony ZYGMUND, On the summability of double Fourier series, Fundamenta Mathematicae 32(1) (1939), 122-132. https://doi.org/10.4064/fm-32-1-122-132
  15. Charles N. MOORE, On convergence factors in double series and double Fourier series, Trans. Amer. Math. Soc. 14 (1913), 73-104.
  16. George E. REVES, Otto SZASZ, Some theorems on double trigonometric series, Duke Math. J. 9 (1942), 693-705. https://doi.org/10.1215/S0012-7094-42-00948-7
  17. P. L. SHARMA, On the harmonic summability of double Fourier series, Annali di Matematica Pura ed Applicata 56(1) (1961), 159-175. https://doi.org/10.1007/BF02414270
  18. P. L. SHARMA, On harmonic summability of double Fourier series, Proc. Amer. Math. Soc. 9 (1958), 979-986. https://doi.org/10.1090/S0002-9939-1958-0104968-8
  19. Per SJOLIN, Convergence almost everywhere of certain singular integrals and multiple Fourier series, Arkiv fur Matematik 9(1-2) (1971), 65-90. https://doi.org/10.1007/BF02383638
  20. N. R. TEVZADZE, The convergence of the double Fourier series at a square summable function, Sakharth. SSR Mecn. Akad. Moambe 58 (1970), 277-279.
  21. Fred USTINA, Convergence of double Fourier series, Annali di Matematica Pura ed Applicata 85(1) (1970), 21-47. https://doi.org/10.1007/BF02413528
  22. Hiroshi WATANABE, Summability of double Fourier series, Tohoku Math. J. 17(2) (1965), 150-160. https://doi.org/10.2748/tmj/1178243581
  23. Anthony J. WHITE, On the restricted Cesaro summability of double Fourier series, Trans. Amer. Math. Soc. 99(2) (1961), 308-319.
  24. Levan V. ZHIZHIASHVILI, On the summation of double Fourier series, Siberian Mathematical Journal 8(3) (1967), 402-414. https://doi.org/10.1007/BF02196423

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  3. 푸리에 급수에 대한 총합가능성의 결과들에 관하여 vol.30, pp.4, 2014, https://doi.org/10.14477/jhm.2017.30.4.233