On the Convergence of Franklin Series to +∞

2019 ◽  
Vol 106 (3-4) ◽  
pp. 334-341 ◽  
Author(s):  
G. G. Gevorkyan
Keyword(s):  
Franklin Series

Uniqueness of Franklin series

1989 ◽  
Vol 46 (2) ◽  
pp. 609-615 ◽  
Author(s):  
G. G. Gevorkyan
Keyword(s):  
Franklin Series

On a “Martingale Property” of Franklin Series

2019 ◽  
Vol 45 (4) ◽  
pp. 803-815
Author(s):  
G. G. Gevorkyan
Keyword(s):  
Martingale Property ◽  
Franklin Series

ON A UNIQUENESS THEOREM FOR THE FRANKLIN SYSTEM

2018 ◽  
Vol 52 (2 (246)) ◽  
pp. 93-100
Author(s):  
K.A. Navasardyan
Keyword(s):  
Uniqueness Theorem ◽  
Partial Sums ◽  
Uniqueness Theorems ◽  
Franklin System ◽  
Almost Everywhere ◽  
Franklin Series

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 $ \lambda \cdot \text{mes} \{ x : \sup\limits_{n}{\left| S_{M_n} (x) \right|} > \lambda \} \to 0 $ as $ \lambda \to +\infty $. 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.

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