APPROXIMATIONS AND THE LINEARITY OF THE SHEPP SPACE

• HONDA Aoi

  Department of Systems Design and Informatics Kyushu Institute of Technology

• OKAZAKI Yoshiaki

  Fuzzy Logic Systems Institute (FLSI)

• SATO Hiroshi

  Faculty of Mathematics Kyushu University

Abstract

Suggested by Shepp [Ann. Math. Statist. 36(4) (1965), 1107-1112] we defined a sequence space Λ_p(f) determined by a single function f(≠ 0) ∈ L_p(R, dx), 1 ≤ p < +∞, and discussed the structure of it. The problems are the linearity and the visible sequential representation of Λ_p(f). In this paper we name Λ<i>_p</i>(<i>f</i>) a Shepp space and discuss the problems in the case of p = 2 by defining an inner approximation Λ⁰₂(f) and an outer approximation Λ^φ₂(f) of Λ₂(f), and we give a necessary and sufficient condition for Λ⁰₂(f) = Λ^φ₂(f) in terms of doubling dimension. In this case Λ₂(f) is a linear space and those approximationsare its visible sequential representations. We also give an example such that Λ₂(f) is a linear space but Λ⁰₂(f) ≠ Λ<i>^φ₂</i>(<i>f</i>).

Journal

• Kyushu Journal of Mathematics

  Kyushu Journal of Mathematics 69 (1), 173-194, 2015

  Faculty of Mathematics, Kyushu University

Keywords

• Shepp space
• linearity
• doubling condition
• doubling dimension
• sequential representation

Details

• CRID: 1390282680206353024
• NII Article ID: 130005076155
• DOI: 10.2206/kyushujm.69.173
• ISSN: 18832032
• ISSN: 13406116
• Web Site: https://www.jstage.jst.go.jp/article/kyushujm/69/1/69_173/_pdf
• Text Lang: en
• Data Source:
  • JaLC
  • Crossref
  • CiNii Articles
  • KAKEN
• Abstract License Flag: Disallowed