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On the mean value of a weakly almost periodic function

Author: L. N. Argabright
Journal: Proc. Amer. Math. Soc. 36 (1972), 315-316
MSC: Primary 43A60
MathSciNet review: 0306820
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Abstract: Let M denote the invariant mean on the space $ W(G)$ of weakly almost periodic functions on a LCA group G. The purpose of this note is to show that, for each $ \phi \in W(G)$,

$\displaystyle M(\phi ) = \mathop {\lim }\limits_{V \to \{ 1\} } \int_G {{{\hat f}_V}(x)\phi (x)dx}$ ($ 1$)

where {V} is the system of compact neighborhoods of 1 in the character group $ \Gamma $, and, for each V, $ {f_V}$ is a continuous positive definite function supported in V and satisfying $ {f_V}(1) = 1$. This answers affirmatively a question recently raised by R. Burckel.

References [Enhancements On Off] (What's this?)

  • [1] R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970. MR 0263963

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Keywords: Weakly almost periodic function, invariant mean
Article copyright: © Copyright 1972 American Mathematical Society

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