The Fekete-Szego problem by a variational method

2021 ◽  
Vol 21 (2) ◽  
pp. 133-150
Author(s):  
Y.V. Borisova ◽  
◽  
I.A. Kolesnikov ◽  
S.A. Kopanev ◽  
G.D. Sadritdinova ◽  
...  
Keyword(s):  
Differential Equation ◽  
Variational Method ◽  
Qualitative Analysis ◽  
Functional Differential Equation ◽  
Complex Analysis ◽  
Classical Method ◽  
The Real ◽  
Sufficient Detail ◽  
Functional Differential ◽  
Boundary Mapping

The article is devoted to the well-known Fekete and Szego problem. The paper investigate the problem in sufficient detail using some new observations by the classical method of internal variations, developed at the Tomsk School of Complex Analysis. One particular case is considered. We carried out complete qualitative analysis of the functional-differential equation relative boundary mapping. We completely solved the problem for the real parameter.

On the approximation of continuous functions by analytic solutions of universal functional-differential equations

2004 ◽  
Vol 41 (1) ◽  
pp. 1-15 ◽  
Author(s):  
C. Elsner
Keyword(s):  
Differential Equation ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Analytic Solutions ◽  
Continuous Functions ◽  
Compact Interval ◽  
The Real ◽  
Real Analytic ◽  
Functional Differential ◽  
Real Analytic Solutions

The existence of an algebraic functional-differential equation P (y′(x), y′(x + log 2), …, y′(x + 5 log 2)) = 0 is proved such that the real-analytic solutions are dense in the space of continuous functions on every compact interval. A similar result holds for an algebraic functional-differential equation P(y′(x − 4πi), y′(x − 2πi), …, y′(x + 4πi)) = 0 (with i2 = −1), which is explicitly given: There are real-analytic solutions on the real line such that every continuous function defined on a compact interval can be approximated by these solutions with arbitrary accuracy.

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