This is the abstract of the paper
  "Banach spaces with the Riemann-Lebesgue or the analytic Riemann-Lebesgue property"
by Shangquan Bu and Ralph Chill.

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Abstract:
In this note we propose and discuss two geometric properties of
Banach spaces related to the validity of the Lemma of
Riemann-Lebesgue in spaces of weakly integrable functions.
We show that the class of Banach spaces with the analytic
Riemann-Lebesgue property is precisely the class of Banach
spaces for which a weak form of a theorem of Ingham holds.

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