Difference between revisions of "Keldysh-Lavrent'ev theorem"
From Encyclopedia of Mathematics
Ulf Rehmann (talk | contribs) m (moved Keldysh–Lavrent'ev theorem to Keldysh-Lavrent'ev theorem: ascii title) |
(TeX) |
||
| Line 1: | Line 1: | ||
| + | {{TEX|done}} | ||
''on uniform approximation by entire functions'' | ''on uniform approximation by entire functions'' | ||
| − | In order that there exist for any continuous complex-valued function | + | In order that there exist for any continuous complex-valued function $f(z)$ on a continuum $E$ and any rapidly-decreasing positive function $\epsilon(r)$, $0\leq r$ (as $r\to\infty$), having a positive lower bound on any finite interval, an entire function $g(z)$ such that |
| − | + | $$|f(z)-g(z)|<\epsilon(|z|),\quad z\in E,$$ | |
| − | it is necessary and sufficient that | + | it is necessary and sufficient that $E$ has no interior points and that there exists a function $\eta(t)$, $0<t<+\infty$, that increases to $+\infty$ and is such that any point $z$ of the complement $CE$ can be joined to $\infty$ by a Jordan curve situated outside $E$ and outside the disc $|\zeta|<\eta(|z|)$. |
This result of M.V. Keldysh and M.A. Lavrent'ev [[#References|[1]]] summarizes numerous investigations on approximation by entire functions initiated by the [[Carleman theorem|Carleman theorem]] (Section 3; see also [[#References|[2]]]). | This result of M.V. Keldysh and M.A. Lavrent'ev [[#References|[1]]] summarizes numerous investigations on approximation by entire functions initiated by the [[Carleman theorem|Carleman theorem]] (Section 3; see also [[#References|[2]]]). | ||
Latest revision as of 18:27, 13 August 2015
on uniform approximation by entire functions
In order that there exist for any continuous complex-valued function $f(z)$ on a continuum $E$ and any rapidly-decreasing positive function $\epsilon(r)$, $0\leq r$ (as $r\to\infty$), having a positive lower bound on any finite interval, an entire function $g(z)$ such that
$$|f(z)-g(z)|<\epsilon(|z|),\quad z\in E,$$
it is necessary and sufficient that $E$ has no interior points and that there exists a function $\eta(t)$, $0<t<+\infty$, that increases to $+\infty$ and is such that any point $z$ of the complement $CE$ can be joined to $\infty$ by a Jordan curve situated outside $E$ and outside the disc $|\zeta|<\eta(|z|)$.
This result of M.V. Keldysh and M.A. Lavrent'ev [1] summarizes numerous investigations on approximation by entire functions initiated by the Carleman theorem (Section 3; see also [2]).
References
| [1] | M.V. Keldysh, M.A. Lavrent'ev, "Sur un problème de M. Carleman" Dokl. Akad. Nauk SSSR , 23 : 8 (1939) pp. 746–748 |
| [2] | S.N. Mergelyan, "Uniform approximation to functions of a complex variable" Transl. Amer. Math. Soc. , 3 (1962) pp. 294–391 Uspekhi Mat. Nauk , 7 : 2 (1952) pp. 31–1A2 |
Comments
References
| [a1] | D. Gaier, "Lectures on complex approximation" , Birkhäuser (1987) (Translated from German) |
How to Cite This Entry:
Keldysh-Lavrent'ev theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Keldysh-Lavrent%27ev_theorem&oldid=36636
Keldysh-Lavrent'ev theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Keldysh-Lavrent%27ev_theorem&oldid=36636
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article