Vol. 15, No. 2, 1965

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Changing signs of Fourier coefficients

Robert E. Edwards

Vol. 15 (1965), No. 2, 463–475

Beginning with a mild extension of a theorem of Littlewood, as generalised by Helgason and by Grothendieck from the circle to a general compact Abelian group G, we derive some properties of the Fourier series of continuous functions on G in relation to arbitrary changes of sign of the coefficients. The main result of this latter type sharpens a fact known for the circle by showing that a continuous function f on G and a ±1-valued function ω on the character group X may be chosen so that

Tωf =    ω(ξ)ˆf(ξ)ξ

belongs to no Orlicz space LΛ(G) for which limu→∞u2Λ(u) = . Similar results are obtained which apply when f is assumed to be merely integrable: in this case one can assert little more than that Tωf is a pseudomeasure on G.

Mathematical Subject Classification
Primary: 42.52
Received: 27 April 1964
Published: 1 June 1965
Robert E. Edwards