Extensions of a Fourier multiplier theorem of Paley

John J. F. Fournier

Vol. 30 (1969), No. 2, 415–431

DOI: 10.2140/pjm.1969.30.415

Abstract

Let A be the class of continuous power series on the unit circle T, that is those continuous functions f whose Fourier coefficients f(n) are 0 for negative indices n. It is known that the most that can be said about the size of the coefficients of such f is that they are square summable. For instance Paley proved the following: Suppose that ∑ ₀^∞|w(n)|^z = ∞. Then there is an f in A with Σ₀^∞|f(n)w(n)| = ∞. In other words the l² sequences are the only multipliers which map A into the class of absolutely convergent power series.

Mathematical Subject Classification

Primary: 42.51

Milestones

Received: 26 January 1968

Published: 1 August 1969