Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지

DIRICHLET-JORDAN THEOREM ON $SIM$ SPACE

  • Kim, Hwa-Joon (Asian Institute of Technology) ;
  • Lekcharoen, S. (Rangsit University) ;
  • Supratid, S. (Rangsit University)
  • Received : 2009.01.06
  • Published : 2009.03.30
Download PDF
⟨ Previous Next ⟩

Abstract

We would like to propose Dirichlet-Jordan theorem on the space of summable in measure(SIM). Surely, this is a kind of extension of bounded variation([1, 4]), and considered as an application of fuzzy set such that ${\alpha}$-cut is 0.

Keywords

References

  1. Donald L. Cohn, Measure theory, (1980), Birkhauser, Boston.
  2. L.A. D'anonio and D. waterman, A summability method for Fourier series of functions of generalized bounded variation, 17 (1997), Analysis, 287-299.
  3. HwaJoon Kim, Some properties of summable in measure, Journal of Applied Mathematics and Computing 25 (2007), 525-531.
  4. P. K. Jain and V. P. Gupta, Lebesgue measure and integration, NCERT, (1982), New Delhi.
  5. Pamela B. Pierce and Daniel Waterman, On the invariance of classes ${\Phi}BV$, ${\Lambda}BV$ under composition, 132 (2003), Proc. Amer. Math. Soc., 755-760.
  6. Pamela B. Pierce and Daniel Waterman, Bounded variation in the mean, 128 (2000), Proc. Amer. Math. Soc., 2593-2596. https://doi.org/10.1090/S0002-9939-00-05391-0
  7. M. Schramm, Functions of $\phi$-bounded variation and Riemann-Stieltjes integration, 287 (1985), Trans. Amer. Math. Soc., 49-63.
  8. M. Schramn and D. Waterman, On the magnitude of Fourier coefficients, 85 (1982), Proc. Amer. Math. Soc., 407-410. https://doi.org/10.1090/S0002-9939-1982-0656113-1
  9. D. Waterman, On ${\Lambda}$-bounded variation, 57 (1976), Studia Math., 33-45. https://doi.org/10.4064/sm-57-1-33-45
  10. D. Waterman, Fourier series of functions of ${\Lambda}$-bounded variation, 74 (1979), Proc. Amer. Math. Soc., 119-123.
  11. L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X