Analysis Mathematica

, Volume 10, Issue 4, pp 301–310 | Cite as

Spectral orders and convolution

  • Yūji Sakai
Article
  • 18 Downloads

Keywords

Spectral Order 

Спектральные порядк и и свертка

Abstract

Известное неравенст во Ф. Рисса (R) ∝ −∞ −∞ fg(y)h(x−y) dx dy ≦ ∝ −∞ −∞ f^(x)g^(y)h^(x−y)dx dy (f,g≧0) переформулирует ся в терминах отношен ия ≺ частичной упорядоче нности ХардиЛиттлву да—Полиа:f×g≺f^×g^(f,g ≧ 0) где f^ и g^ обозначает с имметрические убыва ющие перестановки функци иf и g соответст-венно, аX — операция сверточ ного произведения. До казано, что операция сверточног о произведения сохран яет отношение частич ного порядка в классе неотрицател ьных симметрич-но убывающих функцийf1f2 тогда и только то гда, когдаf 1 ^ ×g≺f2×g дл я всех 0≦g. Даны дальнейшие обоб щения и некоторые при ложения этих теорем.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. C. Allen, Note on a theorem of Gabriel,J. London Math. Soc.,27 (1952), 367–369.Google Scholar
  2. [2]
    T. Bonnesen andW. Fenchel,Theorie der konvexen Körper, Chelsea (New York, 1948).Google Scholar
  3. [3]
    H. J. Brascamp, E. H. Lieb andJ. M. Luttinger, A general rearrangement inequality for multiple integrals,J. Funct. Anal.,17 (1974), 227–237.Google Scholar
  4. [4]
    H. J. Brascamp andE. H. Lieb, Best constant in Young's inequality, its converse, and its generalization to more than three functions,Adv. in Math.,20 (1976), 151–173.Google Scholar
  5. [5]
    K. M.Chong and N. M.Rice, Equimeasurable rearrangements of functions,Queen's Papers in Pure and Applied Math., No.28, 1971.Google Scholar
  6. [6]
    K. M. Chong, Some extentions of a theorem of Hardy, Littlewood and Pólya and their applications,Canad. J. Math.,26 (1974), 1321–1340.Google Scholar
  7. [7]
    P. W. Day, Decreasing rearrangements and doubly stochastic operators,Trans. Amer. Math. Soc.,178 (1973), 383–392.Google Scholar
  8. [8]
    R. Friedberg andJ. M. Luttinger, Rearrangement inequality for periodic functions,Arch. Rational Mech. Anal.,61 (1976), 35–44.Google Scholar
  9. [9]
    R. M. Gabriel, A “star inequality” for harmonic functions,Proc. London Math. Soc. (2),34 (1931), 305–313.Google Scholar
  10. [10]
    G. H. Hardy, J. E. Littlewood andG. Pólya,Inequalities, second ed., Cambridge Univ. Press (London-New York-Melbourne, 1978).Google Scholar
  11. [11]
    L. Leindler, On a certain converse of Hölder's inequality. II,Acta Sci. Math. Szeged,33 (1972), 217–223.Google Scholar
  12. [12]
    G. G. Lorentz,Bernstein Polynomials, Toronto Univ. Press (Toronto, 1953).Google Scholar
  13. [13]
    J. M. Luttinger andR. Friedberg, A new rearrangement inequality for multiple integrals,Arch. Rational Mech. Anal.,61 (1976), 45–64.Google Scholar
  14. [14]
    W. A. J. Luxemburg, Rearrangement invariant Banach function spaces,Queen's Papers in Pure and Applied Math., No.10 (1967), 83–144.Google Scholar
  15. [15]
    F. Riesz, Sur une inégalité intégrale,J. London Math. Soc.,5 (1930), 162–168.Google Scholar
  16. [16]
    Y.Sakai, Weak spectral order of Hardy, Littlewood and Pólya,J. Math. Anal. Appl., to appear.Google Scholar
  17. [17]
    Y.Sakai, Strong spectral order of Hardy, Littlewood and Pôlya, submitted.Google Scholar
  18. [18]
    C. E. Shannon andW. Weaver,The mathematical theory of communications, The University of Illinois Press (Urbana, 1964).Google Scholar

Copyright information

© Akadämiai Kiadó 1984

Authors and Affiliations

  • Yūji Sakai
    • 1
  1. 1.Department of Applied Mathematics Faculty of EngineeringShinshu UniversityNagano-ShiJapan

Personalised recommendations