Classification of Distributions by the Arithmetic Means of Their Fourier Series

Abstract

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 \(\sum {c_n } e^{inx} \) 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 u e Cm; 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 F e Dm−1, which is not true in general if F e Dm.