Divergence of decreasing rearranged Fourier series
In the present paper, the author shows that there exists a real function \(f\in L^2(\mathbb{T})\) such that \[ \limsup_{\eta\to 0+} \Biggl| \sum_{|\widehat f(k)|\geq \eta} \widehat f(k)\exp(2\pi ikt)\Biggr|= \infty \] for almost all \(t\in [0,1)\), where \(\widehat f(k)\) is the \(k\)th Fourier coefficient of \(f\). This result strengthens the result on the rearrangement of Fourier series stated by Kolmogorov and proven by Ul'yanov and later much simplified by Olevskij: There exists a real \(f\in L^2(\mathbb{T})\) and a bijection \(\sigma:{\mathbf Z}\to{\mathbf Z}\) such that \[ \limsup_{N\to\infty} \Biggl| \sum_{| k|\leq N}\widehat f(\sigma(k))\exp(2\pi i\sigma(k)t)\Biggr|= \infty \] for almost every \(t\in [0,1)\). To construct the function, the author modifies the proof of Olevskij. The concrete construction is rather involved. Using a probabilistic argument basic building blocks are produced finally to construct a modified Olevskij system.
scientific article; zbMATH DE number 935305