||APPROXIMATIONS AND THE LINEARITY OF THE SHEPP SPACE
Honda, Aoi ,
Okazaki, YoshiakiSato, Hiroshi
Kyushu Journal of Mathematics
194 , 2015-06-09 , Faculty of Mathematics, Kyushu University
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_p(f) a Shepp space and discuss the problems in the case of p = 2 by defining an inner approximation Λ^0_2(f) and an outer approximation Λ^φ_2(f) of Λ_2 (f), and we give a necessary and sufficient condition for Λ^0_2(f) =Λ^φ_2 (f) in terms of doubling dimension. In this case Λ_2(f) is a linear space and those approximationsare its visible sequential representations. We also give an example such that Λ_2(f) is a linear space butΛ^0_2 (f) ≠Λ^φ_2.