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 · mes { x : _n| S_M_n (x) | > λ } → 0 as λ → +∞ . This shows that the boundedness assumption of the ratio M_n+1 M_n , used for the proofs of uniqueness theorems in earlier papers, can not be omitted.