Classification of Distributions by the Arithmetic Means of Their Fourier Series

Author: Kádár, F.

Source: Acta Mathematica Hungarica, Volume 89, Number 3, November 2000 , pp. 221-232(12)

Publisher: Springer

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Abstract:

We consider the class C^m of functions that are m times differentiable on the one-dimensional torus group G = R/2Z with respect to addition mod 2; 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 $$\sum {c_ne^{inx}}$$ is the Fourier series of a distribution in D^m if and only if the sequence of its arithmetic means _N (u) as distributions is bounded for all u ∈ C; or equivalently, if sup ‖_N¦_D < ∞. Another result says that the arithmetic mean _N F of a distribution converges to F in the strong topology of D^m if F ∈ D^m−1, which is not true in general if F ∈ D^m.

Document Type: Regular Paper

Affiliations: Vág Utca 5 1133 Budapest Hungary E-mail: FERI@MARECO.HU

Publication date: November 1, 2000

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