On Fourier series that are universal modulo signs
Volume 249 / 2019
Studia Mathematica 249 (2019), 215-231
MSC: Primary 42A16; Secondary 42B05.
DOI: 10.4064/sm180628-15-10
Published online: 13 May 2019
Abstract
A function $U\in L^{1}(0,2\pi)$ and a unit density subset $\mathcal{S}$ of the set of positive integers are constructed with the following property: for each measurable function $f$ on $[0,2\pi]$ one can find a sequence $\{\delta_{k}=\pm1\}_{k=1}^{\infty}$ such that \begin{equation*} \lim_{n\in \mathcal{S},\, n \rightarrow\infty}\sum_{\nu=1}^{n}\delta_{\nu} (a_{\nu}(U)\cos \nu x+b_{\nu}(U)\sin \nu x )=f(x)\quad\ \text{a.e. on } (0,2\pi), \end{equation*} where \begin{equation*} a_{\nu}(U)=\frac{1}{\pi}\int_{0}^{2\pi} U(t)\cos\nu t\,dt, \quad b_{\nu}(U)=\frac{1}{\pi}\int_{0}^{2\pi} U(t)\sin\nu t\,dt\quad \ (\nu=1, 2, \ldots) \end{equation*} are the Fourier coefficients of $U$.
