APPROXIMATELY CONVEX SCHWARTZ DISTRIBUTIONS

  • Published : 2009.05.31

Abstract

Generalizing the approximately convex function which is introduced by D.H. Hyers and S.M. Ulam we establish an approximately convex Schwartz distribution and prove that every approximately convex Schwartz distribution is an approximately convex function.

Keywords

References

  1. S. Banach: Theorie des Operationes Lineaires. Warsaw, 1932.
  2. P. W. Cholewa: Remarks on the stability of functional equations. Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  3. E. Deeba, P. K. Sahoo & Shishen Xie: On a class of functional equations in distributions. J. Math. Anal. Appl. 223 (1998), 334-346. https://doi.org/10.1006/jmaa.1998.5995
  4. E. Deeba & Shishen Xie: Distributional analog of a functional equation. Applied Mathematics Letters 16 (2003), 669-673. https://doi.org/10.1016/S0893-9659(03)00065-X
  5. I. M. Gelfand & G. E. Shilov: Generalized functions IV, Academic. Press, New York, 1968.
  6. J. W. Green: Approximately convex functions. Duke Math. J. 19 (1952a), 499-504. https://doi.org/10.1215/S0012-7094-52-01952-2
  7. L. Hormander: The Analysis of Linear Partial Differential Operator I. Springer-Verlag, Berlin-New York, 1983.
  8. D. H. Hyers & S. M. Ulam: Approximately convex functions. Proc. Amer. Math. Soc. 3 (1954), 821-828. https://doi.org/10.1090/S0002-9939-1952-0049962-5
  9. D. H. Hyers, G. Isac & Th. M. Rassias: Stability of Functional Equations in Several Variables. Birkhauser, 1998.
  10. L. Schwartz: Theorie des distributions. I, II, 2nd ed. Hermann, Paris, 1957.