A characterization of the exponential distribution was obtained by Grosswald et al. (1980) using the relevation transform introduced by Krakowski (1973). Here we obtain an improved version of the result in Grosswald et al. (1980).
Wiener defined an integrated Fourier transformation and proved that this transformation is an isometry from the nonlinear subspace '5liV(R) of %R2(R) consisting of functions of bounded average quadratic power, into the nonlinear subspace GW(R) of V(IR) consisting of functions of bounded quadratic variation. By using two generalized Tauberian theorems, we prove that Wiener's transformation W is actually an isomorphism from .)2(R) onto 'V2(R). We also show by counterexamples that W is not an isometry on the closed subspace generated by 6V2(R).1. Introduction. The purpose of this paper is to find out how Wiener's generalized harmonic analysis [18] fits into the framework of contemporary functional analysis.
Abstract. A closed subspace M in a Banach space X is called t/-proximinal if it satisfies: (1 + p)S n (S + M) ç S + e(pXS n M), for some positive valued function t(p), p > 0, and e(p) -» 0 as p -> 0, where 5 is the closed unit ball of X. One of the important properties of this class of subspaces is that the metric projections are continuous. We show that many interesting subspaces are (/-proximinal, for example, the subspaces with the 2-ball property (semi M-ideals) and certain subspaces of compact operators in the spaces of bounded linear operators.1. Introduction. We call a closed subspace M of a real Banach space X an M-ideal if the annihilator M ± of M is an L-summand in X*. This notion was formulated and studied by Alfsen and Effros [1]. It was proved that if M is an A/-ideal, then M is a proximinal subspace of X [1], [5]. In [4], Hennefeld showed that the space of compact operators on F (or c0), 1 < p < oo, is an M-ideal in the space of bounded linear operators on F (or c0 respectively). This theorem is also true for operators from F into lq, 1 < p < q < oo [11]. (It is well known that if 1 < q < p < oo, then every bounded linear operator from /' into I9 is compact.) Af-ideal theory provides a convenient tool to study the approximation of operators by the space of compact operators and
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