Abstract
We show by example that the classical characterization of the Fourier series of periodic functions in Lp, 1<p≤+∞, as those trigonometric series whose Abel or Fejer means are uniformly bounded in Lp does not hold for general (non-periodic) trigonometric series in relation to Stepanov-almost-periodic functions, but that it does hold under the additional hypothesis that the means are translation equicontinuous. We exhibit a bounded, infinitely differentiable function that belongs to every class of Besicovitch-almost-periodic functions but is not equivalent in the metric of Besicovitch-almost-periodic functions to any Stepanov-almost-periodic function.
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