# RELATIONS BETWEEN TWO CLASSES OF FUNCTIONS

Document Type : Research Paper

Author

Department of Mathematics, University of hormozgan, Bandarabbas, Iran

Abstract

Let F denote a specific space of the class of was costructed by H. Khodabakhshian
as a classes of separable Banach function spaces analogous to the james function spaces. In this
notes we prove that l_p(α) is isomorphic to a complemented subspace of F_{α,p} and for p = 2, F_{α,p} is a closed subspace of the Waterman-Shiba space αBV^ (p)
Assume F denotes a specific space of the class of F_{α,p} that was costructed by H.
Khodabakhshian[2] as a classes of separable Banach function spaces analogous to the James
function spaces. In this notes we prove that l_p(α) is isomorphic to a complemented subspace of
F_{α,p} and for p = 2, F_{α,p} is a closed subspace of Waterman-Shiba space αBV^(p).

Keywords

#### References

[1] S. Buechler, Ph.D. Thesis, University of Texas at Austin, 1994.
[2] H. Khodabakhshian, Ph.D. Thesis, University of Sistan and Baluchestan, 2008.
[3] P. Azimi, J. Hagler, Example of hereditarily l1 Banach spaces failing the Scher property,
Paci c J. Math. 122 (1986), 287-297.
[4] M. M. Day, Normal Linear Spaces, Springer Verlag, Berlin, 1958.
[5] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Vol.I sequences spaces,
Springer Verlag, Berlin. 1996.
[6] M. Shiba, On the absolute convergence of Fourier series of functions of class BV (p),
Sci. Rep. Fukushima Univ. 30 (1980), 7-10.
[7] D. Waterman, On convergence of Fourier series of functions of bounded generalized
variation, Studia Math. 44 (1972), 107-117.
[8] D. Waterman, On -bounded variation, Studia Math. 57 (1976), 33-45.

### History

• Receive Date: 25 August 2019
• Revise Date: 14 February 2021
• Accept Date: 06 April 2021
• First Publish Date: 01 May 2021