ON A UNIQUENESS THEOREM FOR THE FRANKLIN SYSTEM - Navasardyan, K.A. (Proceedings of the YSU A: Physical and Mathematical Sciences, 2018)
In this paper we prove that there exist a nontrivial Franklin series and a sequence $ M_n $ such that the partial sums $ S_{M_n} (x) $ of that series converge to 0 almost everywhere and $ λ · \text{mes} \{ x : \sup\limits_{n}{\left| S_{M_n} (x) \right|} > λ \} → 0 $ as $ λ → +∞ $. This shows that the boundedness assumption of the ratio $ \dfrac{ M_{n+1}}{M_n} $, used for the proofs of uniqueness theorems in earlier papers, can not be omitted.
Keywords: majorant of partial sums; Franklin system; uniqueness