Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 101, Number 2, October 1987
The purpose of this department is to publish very short papers of unusually
polished character, for which there is no other outlet.
FOURIER SERIES WITH POSITIVE COEFFICIENTS
J. MARSHALL ASH,1 MICHAEL RAINS AND STEPHEN VAGI
ABSTRACT. Extending a result of N. Wiener, it is shown that functions on
the circle with positive Fourier coefficients that are pth power integrable near
0, 1 < p < 2, have Fourier coefficients in 1P".
The following result was proved (but never published) by Norbert Wiener in the
early 1950's. (See [1, pp. 242, 250] and .)
WIENER'S THEOREM. If E Cneint is the Fourier series of a function f E
L1(-ir,ir) with Cn > 0 for all n, and f restricted to a neighborhood(-6,6) of
the origin belong to L2(-6, 6), then f belongs to L2(-_r, ir).
A question which immediately arises in connection with this result is the fol-
lowing: does the theorem remain true if one replaces L2(-6,6) and L2 (-_r,ir) in
its statement respectively by LP(-6, 6) and LP(--x, x), with 1 < p < ox? In 1969
Stephen Wainger showed, by ingenious counterexamples, that the answer is neg-