Classification of Distributions by the Arithmetic Means of Their Fourier Series

Abstract

We consider the class C ^m 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 D ^m of distributions of order at most m. Clearly, D ^m can be identified as the dual space of C ^m. One of our main results says that a formal trigonometric series $\sum {c_n } e^{inx}$ is the Fourier series of a distribution in D ^m if and only if the sequence of its arithmetic means σ _N (u) as distributions is bounded for all u ε C ^m; or equivalently, if sup ‖σ _N ¦ _{D m} < ∞. Another result says that the arithmetic mean σ _N F of a distribution converges to F in the strong topology of D ^m if F ε D ^m−1, which is not true in general if F ε D ^m.