Divergence of Fourier-Haar series of bounded functions on sets of measure zero
Answering a question posed by the reviewer [Colloq. Math. 52, 213-238 (1987; Zbl 0642.42024)] the author proves that given a set \(E \subseteq [0,1]\) of Lebesgue measure zero, there is a bounded measurable function \(f\) on \([0,1]\) whose Haar-Fourier series diverges everywhere on \(E\). The construction is based on a clever iterative procedure which generates a sequence of nested \(G_ \delta\) sets (whose components are dyadic intervals) which approximate \(E\). These sets are used to define \(f\) and to control the size of each term of the Haar-Fourier series of \(f\) at points on \(E\).
scientific article; zbMATH DE number 525666