APPROXIMATIONS AND THE LINEARITY OF THE SHEPP SPACE

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Suggested by Shepp [Ann. Math. Statist. 36(4) (1965), 1107-1112] we defined a sequence space Λp(f) determined by a single function f(≠ 0) ∈ Lp(R, dx), 1 ≤ p < +∞, and discussed the structure of it. The problems are the linearity and the visible sequential representation of Λp(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 Λφ2(f) of Λ2(f), and we give a necessary and sufficient condition for Λ02(f) = Λφ2(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>).

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