Classification of Distributions by the Arithmetic Means of Their Fourier Series

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We consider the class Cm of functions that are m times differentiable on the one-dimensional torus group G = R/2πZ with respect to addition mod 2π; and the class of Dm of distributions of order at most m. Clearly, Dm can be identified as the dual space of Cm. One of our main results says that a formal trigonometric series ∑cneinx is the Fourier series of a distribution in Dm if and only if the sequence of its arithmetic means σN (u) as distributions is bounded for all uCm; or equivalently, if sup ‖σN¦Dm < ∞. Another result says that the arithmetic mean σNF of a distribution converges to F in the strong topology of Dm if FDm−1, which is not true in general if FDm.

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