On the representation of functions by trigonometric series
The aim of this fascinating paper is to give an account of a number of Menshov's theorems among others, on the adjustment of measurable functions, on the existence of universal trigonometric series, on the limits of indeterminacy of trigonometric series, etc. The present author has developed an approach different from that of Menshov and presents a coherent treatment in the paper under review. Let us write \[ S_n(f,x):= \sum^n_{k=-n}\widehat f(k)e^{ikx} \] and, if \(E\) is a measurable set, let us write \(|E|\) for the Lebesgue measure of \(E\) normalized such that \(|\mathbb{T}|=1\). The key result on which everything else depends is formulated in the following lemma. Given any \(\eta>0\) there exists a \(K(\eta)\) with the following property. Given any \(\varepsilon>0\) one can find a positive function \(f\in A(\mathbb{T})\), that is \(\sum|\widehat f(k)|<\infty\), with the following properties: (i) \(\widehat f(0)=1\), (ii) \(|\widehat f(k)|\leq\varepsilon\) for all \(k\neq 0\), (iii) \(|S_m(f,x)|\leq K(\eta)\) for all \(m\) and \(x\), (iv) \(|\{x:f(x)\neq 0\}|<\eta\). We warmly recommend this paper to everyone who is interested in harmonic analysis.
scientific article; zbMATH DE number 1013858
scientific article