Gap series and an example to Malliavin’s theorem

Robert P. Kaufman

Vol. 28 (1969), No. 1, 117–119

DOI: 10.2140/pjm.1969.28.117

Abstract

O. Malliavin’s celebrated theorem of spectral nonsynthesis is based on a real function f of class A

f(t) =	∑ n=1∞a n cosnt + ∑ n=1∞b n sinnt,
	∑ |an| + ∑ |bn| < ∞,

for which ∫ _−∞^∞|u|∥e^iuf∥_∞du < ∞.

Here and in general ∥g∥_∞≡ sup_n|ĝ(n)|. This note presents a method for constructing a function f, based on a gap property and a method of estimation of Kahane.

Mathematical Subject Classification

Primary: 42.58

Milestones

Received: 9 October 1967

Published: 1 January 1969