DOI QR코드

DOI QR Code

On Lp(T2)-Convergence and Móricz

Lp(T2)-수렴성과 모리츠에 관하여

  • LEE, Jung Oh (Dept. of liberal arts, Chosun College of Science and Technology)
  • Received : 2015.10.27
  • Accepted : 2015.12.11
  • Published : 2015.12.31

Abstract

This paper is concerned with the convergence of double trigonometric series and Fourier series. Since the beginning of the 20th century, many authors have studied on those series. Also, Ferenc $M{\acute{o}}ricz$ has studied the convergence of double trigonometric series and double Fourier series so far. We consider $L^p(T^2)$-convergence results focused on the Ferenc $M{\acute{o}}ricz^{\prime}s$ studies from the second half of the 20th century up to now. In section 2, we reintroduce some of Ferenc $M{\acute{o}}ricz^{\prime}s$ remarkable theorems. Also we investigate his several important results. In conclusion, we investigate his research trends and the simple minor genealogy from J. B. Joseph Fourier to Ferenc $M{\acute{o}}ricz$. In addition, we present the research minor lineage of his study on $L^p(T^2)$-convergence.

References

  1. Chang-Pao CHEN, Hui-Chuan WU, F. MORICZ, Pointwise convergence of multiple trigonometric series., J. Math. Anal. Appl. 185(3) (1994), 629-646. https://doi.org/10.1006/jmaa.1994.1273
  2. X. Z. KRASNIQI, P. KORUS, F. MORICZ, Necessary conditions for the $L^p$-convergence (0 < p < 1) of single and double trigonometric series., Mathematica Bohemica 139(1) (2014), 75-88.
  3. L. KRIZSAN, F. MORICZ, The Lebesque summability of double triginometric integrals, Mathematical Inequalities & Applications 17(4) (2014), 1543-1550.
  4. LEE Jung Oh , A brief study on Bhatia's research of $L^1$-convergence, The Korean Journal for History of Mathematics 27(1) (2014), 81-93. https://doi.org/10.14477/jhm.2014.27.1.081
  5. LEE Jung Oh, On Classical Studies for the Summability and Convergence of Double Fourier Series, The Korean Journal for History of Mathematics, 27(4) (2014), 285-297. https://doi.org/10.14477/jhm.2014.27.4.285
  6. F. MORICZ, Convergence and integrability of double trigonometric series with coefficients of bounded variation, Proc. Am. Math. Soc.102(3) (1988), 633-640. https://doi.org/10.1090/S0002-9939-1988-0928995-2
  7. F. MORICZ, On the integrability of double cosine and sine series. I., J. Math. Anal. Appl. 154(2) (1991), 452-465. https://doi.org/10.1016/0022-247X(91)90050-A
  8. F. MORICZ, On the integrability of double cosine and sine series. II., J. Math. Anal. Appl. 154(2) (1991), 466-483. https://doi.org/10.1016/0022-247X(91)90051-Z
  9. F. MORICZ, $L^1$-convergence of double Fourier series., Journal of Mathematical Analysis and Applications 186 (1994), 209-236. https://doi.org/10.1006/jmaa.1994.1295
  10. F. MORICZ, On the maximal Fejer operator for double Fourier series of functions in Hardy spaces., Stud. Math. 116(1) (1995), 89-100. https://doi.org/10.4064/sm-116-1-89-100
  11. F. MORICZ, Necessary conditions for $L^1$-convergence of double Fourier series, J. Math. Anal. Appl. 363 (2010), 559-568. https://doi.org/10.1016/j.jmaa.2009.09.030
  12. F. MORICZ, M. BAGOTA, On the Lebesgue summability of double trigonometric series, J. Math. Anal. Appl. 348 (2008), 555-561. https://doi.org/10.1016/j.jmaa.2008.07.064
  13. F. MORICZ, B. E. RHOADES, Approximation by Norlund means of double Fourier series for Lipschitz functions, J. Approximation Theory 50 (1987), 341-358. https://doi.org/10.1016/0021-9045(87)90012-8
  14. F. MORICZ, Shi, Xianliang, Approximation to continuous functions by Cesaro means of double Fourier series and conjugate series, J. Approximation Theory 49 (1987), 346-377. https://doi.org/10.1016/0021-9045(87)90074-8
  15. F. MORICZ, Daniel WATERMAN, Convergence of double Fourier series with coefficients of generalized bounded variation., J. Math. Anal. Appl. 140(1) (1989), 34-49. https://doi.org/10.1016/0022-247X(89)90092-9