Uniqueness Theorems for Franklin Series

2018 ◽  
Vol 303 (1) ◽  
pp. 58-77 ◽  
Author(s):  
G. G. Gevorkyan
Keyword(s):  
Uniqueness Theorems ◽  
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.

Uniqueness Theorems for Multiple Franklin Series Converging over Rectangles

2021 ◽  
Vol 109 (1-2) ◽  
pp. 208-217
Author(s):  
G. G. Gevorkyan ◽  
L. A. Hakobyan
Keyword(s):  
Uniqueness Theorems ◽  
Franklin Series
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