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Mathematical Notes

, Volume 52, Issue 3, pp 943–947 | Cite as

Equivalent criterion of Haar and Franklin systems in symmetric spaces

  • I. Ya. Novikov
Article
  • 33 Downloads

Abstract

In the present article it is proved that if the Haar and Franklin systems are equivalent in a separable symmetric space E, the following condition holds: 0<αE⩽βE<1, (1) where αE and βE are the Boyd indices of the space E. It is already known that if condition (1) is fulfilled, it follows that the Haar and Franklin systems are equivalent in the space E. Thereby, this estabishes that condition (1) is necessary and sufficient for the equivalence of the Haar and Franklin systems in E.

In proving the assertion a number of interesting constructions involving Haar and Franklin polynomials are presented and upper and lower bounds on the Franklin functions applied.

Keywords

Lower Bound Present Article Symmetric Space Equivalent Criterion Franklin System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature cited

  1. 1.
    B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Nauka, Moscow (1984).Google Scholar
  2. 2.
    Z. Ciesielski, P. Simon, and P. Sjölin, “Equivalence of Haar and Franklin bases in Lp spaces,” Studia Math.,60, No. 2, 195–211 (1976).Google Scholar
  3. 3.
    S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978).Google Scholar
  4. 4.
    A. A. Komissarov, “On the equivalence of the Haar and Franklin systems in symmetric spaces,” Usp. Mat. Nauk,2, No. 2, 203–204 (1982).Google Scholar
  5. 5.
    A. A. Komissarov, “On the equivalence of the Haar and Franklin systems in certain function spaces,” Sib. Mat. Zh.,23, No. 5, 115–126 (1982).Google Scholar
  6. 6.
    J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. Function Spaces, Springer, Berlin (1979).Google Scholar
  7. 7.
    Z. Ciesielski, “Properties of the orthonormal Franklin system. I, II,” Studia Math.,23, No. 2, 141–157 (1963); ibid.,27, No. 3, 289–323 (1966).Google Scholar
  8. 8.
    P. Sjölin, “The Haar and Franklin systems are not equivalent in L1,” Bull. Acad. Pol. Sci. Ser. Sci., Math., Astron. Phys.25, No. 11, 1099–1100 (1977).Google Scholar
  9. 9.
    T. Shimogaki, “A note on norms of compression operators of function spaces,” Proc. Japan. Acad.,46, No. 3, 239–292 (1970).Google Scholar
  10. 10.
    S. V. Astashkin, “Example of a symmetric space not of fundamental type,” in: Proc. 14th School on Operator Theory in Functional Spaces, NGU, Novgorod (1989). Part I, p. 19.Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • I. Ya. Novikov
    • 1
  1. 1.Mathematics Research InstituteVoronezh State UniversityUSSR

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