A Uniqueness Theorem for Franklin Series

2020 ◽  
Vol 55 (3) ◽  
pp. 166-178
Author(s):  
K. Keryan ◽  
A. Khachatryan
Keyword(s):  
Uniqueness Theorem ◽  
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 theorem for multiple Franklin series

2017 ◽  
Vol 101 (1-2) ◽  
pp. 219-229 ◽  
Author(s):  
G. G. Gevorkyan
Keyword(s):  
Uniqueness Theorem ◽  
Franklin Series

Teichmuller Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Uniqueness Theorem ◽  
Existence Theorems ◽  
Quadratic Differentials ◽  
Complex Structures ◽  
Metric Geometry ◽  
Quasiconformal Maps ◽  
Hyperbolic Structures ◽  
Teichmüller Metric ◽  
Uniqueness And Existence ◽  
Teichmüller Mapping

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

scholarly journals Havin-Mazya type uniqueness Theorem for Dirichlet spaces

2021 ◽  
pp. 102967
Author(s):  
H. Bahajji-El Idrissi ◽  
O. El-Fallah ◽  
K. Kellay
Keyword(s):  
Uniqueness Theorem ◽  
Dirichlet Spaces
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