On uniqueness for series in the general Franklin system

Abstract

We prove some uniqueness theorems for series in general Franklin systems. In particular, for series in the classical Franklin system our result asserts that if the partial sums $S_{n_i}(x)=\sum_{k=0}^{n_i}a_kf_k(x)$ of a Franklin series $\sum_{k=0}^{\infty}a_kf_k(x)$ converge in measure to an integrable function $f$ and $\sup_i|S_{n_i}(x)|<\infty$, for $x\notin B$, where $B$ is some countable set and $\sup_i(n_i/n_{i-1})<\infty$, then this is the Fourier-Franklin series of $f$. Bibliography: 29 titles.