On a method of approximation by gradients

Author:
M. B. Dubashinskiĭ

Translated by:
the author

Original publication:
Algebra i Analiz, tom **25** (2013), nomer 1.

Journal:
St. Petersburg Math. J. **25** (2014), 1-22

MSC (2010):
Primary 41A30, 41A63

DOI:
https://doi.org/10.1090/S1061-0022-2013-01277-5

Published electronically:
November 20, 2013

MathSciNet review:
3113426

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Abstract | References | Similar Articles | Additional Information

Abstract: The subject of this study is the possibility of approximation of a continuous vector field on a compact set by gradients of smooth functions defined on the entire . A method is obtained that yields either approximation or an obstruction for it. This method does not involve the Hahn-Banach theorem and is based on solving a quasilinear elliptic equation in partial derivatives. A discrete analog of the above problem is studied, namely, the problem of approximation by gradients on a finite oriented graph. A stepwise algorithm is suggested in this case.

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Additional Information

**M. B. Dubashinskiĭ**

Affiliation:
Chebyshev Laboratory, St. Petersburg State University, 14th Line 29B, Vasilyevsky Island, St. Petersburg 199178, Russia

Email:
mikhail.dubashinskiy@gmail.com

DOI:
https://doi.org/10.1090/S1061-0022-2013-01277-5

Keywords:
Approximation by gradients,
solenoidal vector charge,
direct methods,
analysis on graphs

Received by editor(s):
September 30, 2012

Published electronically:
November 20, 2013

Additional Notes:
The author was supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government, grant 11.G34.31.0026

Article copyright:
© Copyright 2013
American Mathematical Society