In this paper, we discuss the linearity of a sequence space $\Lambda_{p}(f)$, and the conditions such that $\ell_{1} = \Lambda_{1}(f)$ holds are characterized in term of the essential bounded variation of $f\in L_{1}(\mathbf{R})$, i.e. $\ell_{1} = \Lambda_{1}(f)$ if and only if $f\in BV(\mathbf{R})$.
References
A. Honda, Y. Okazaki and H. Sato, An $L_{p}$-function determines $l_{p}$, Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 3, 39–41. MR2398577 10.3792/pjaa.84.39 euclid.pja/1204555683
A. Honda, Y. Okazaki and H. Sato, An $L_{p}$-function determines $l_{p}$, Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 3, 39–41. MR2398577 10.3792/pjaa.84.39 euclid.pja/1204555683
A. Honda, Y. Okazaki and H. Sato, A new sequence space defined by an $L_{2}$-function, in Banach and Function Spaces III (held at Kyushu Institute of Technology (KIT), Tobata Campus, Kitakyushu, JAPAN on September 14–17, 2009), Proceedings of the Third International Symposium on Banach and Function Spaces 2009, Yokohama Publishers, Yokohama. (to appear).A. Honda, Y. Okazaki and H. Sato, A new sequence space defined by an $L_{2}$-function, in Banach and Function Spaces III (held at Kyushu Institute of Technology (KIT), Tobata Campus, Kitakyushu, JAPAN on September 14–17, 2009), Proceedings of the Third International Symposium on Banach and Function Spaces 2009, Yokohama Publishers, Yokohama. (to appear).
L. A. Shepp, Distingunishing a sequence of random variables from a translate of itself, Ann. Math. Statist. 36 (1965), 1107–1112. MR176509 10.1214/aoms/1177699985 euclid.aoms/1177699985
L. A. Shepp, Distingunishing a sequence of random variables from a translate of itself, Ann. Math. Statist. 36 (1965), 1107–1112. MR176509 10.1214/aoms/1177699985 euclid.aoms/1177699985
