Vol. 30, No. 2, 1969

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Extensions of a Fourier multiplier theorem of Paley

John J. F. Fournier

Vol. 30 (1969), No. 2, 415–431

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
Received: 26 January 1968
Published: 1 August 1969
John J. F. Fournier