View source for Approximation of functions of a complex variable

The branch of complex analysis studying problems regarding the approximate representation (approximation) of functions of a complex variable by means of analytic functions of a specific class. The fundamental problems in the theory of approximation of functions of a complex variable are: the possibility of approximating; the rate of approximation; and the approximation properties of various methods of representing functions (interpolation sequences and series, series in orthogonal polynomials or Faber polynomials, expansion in continued fractions and Padé approximation, sequences of polynomials in exponential functions, Dirichlet series, etc.). The theory of approximation of functions of a complex variable is intimately connected with other branches of complex analysis, and with mathematics in general. Methods and results on conformal mapping, integral representation, potential theory, the theory of function algebras, etc., play an important role in approximation theory.

The central problems in the theory of approximation of functions of a complex variable relate to approximation by polynomials or rational functions, in particular by polynomials and rational functions of best approximation (existence, characteristic properties, uniqueness), as well as to extremal problems and various estimates for polynomials and rational functions (growth estimates, inequalities for derivatives, polynomials and rational functions, least deviation from zero, etc.).

''A.A. Gonchar''

==The approximation of functions of a complex variable by polynomials and rational functions.==
In this branch of approximation theory one may distinguish several directions.

1) The study of the possibility of approximating a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a0130001.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a0130002.png" /> with given accuracy by polynomials or rational functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a0130003.png" />, in dependence on the properties of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a0130004.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a0130005.png" /> is given and on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a0130006.png" /> is to be approximated, on the properties of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a0130007.png" /> of deviation and on the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a0130008.png" /> itself.

2) The study of the properties of polynomials and rational functions of best approximation, i.e. polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a0130009.png" /> and rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300010.png" /> of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300012.png" /> for which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300013.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300014.png" /></td> </tr></table>

where the infima are over the set of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300015.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300016.png" /> or rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300017.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300018.png" />, respectively (or over parts of these sets, distinguished by addition requirements). In fact, one deals here with properties of solutions for a certain class of extremal problems. In this context one may also consider the study of other extremal problems on sets of polynomials, rational functions and on certain classes of analytic functions, as well as the study of analytic properties of polynomials and rational functions (in particular, obtaining inequalities between various norms of these functions and their derivatives).

3) The study of the dependence of the rate of decrease (to zero) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300020.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300021.png" />, on the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300024.png" /> (the so-called direct theorems of approximation theory) and the dependence of the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300025.png" /> on the rate of decrease of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300027.png" /> to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300028.png" /> and on the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300030.png" /> (inverse theorems). The study of approximation properties of well-known methods in approximation theory (e.g., series in [[Faber polynomials|Faber polynomials]], various interpolation processes (cf. [[Interpolation process|Interpolation process]])), as well as the search for new effective approximation methods are related to this direction.

4) The approximation of functions of several complex variables. Here, basically, the same problems are solved as in the case of one complex variable, but the results and the methods for obtaining them differ, as a rule, sharply from those used in the case of one variable.

In the following some basic results have been listed.

1) The problem of the existence of a uniform approximation by polynomials that is as good as one pleases is solved by the [[Runge theorem|Runge theorem]] (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300031.png" /> is analytic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300032.png" />), the [[Lavrent'ev theorem|Lavrent'ev theorem]] (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300033.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300034.png" />), the [[Keldysh theorem|Keldysh theorem]] (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300035.png" /> is a closed domain, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300036.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300037.png" /> and analytic within <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300038.png" />) and the [[Mergelyan theorem|Mergelyan theorem]] (in the general case: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300039.png" /> is a compact set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300040.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300041.png" /> and analytic at interior points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300042.png" />).

2) The problem whether it is possible to approximate holomorphic functions on closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300043.png" /> in the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300044.png" /> is solved by Runge's theorem. In the study of approximating a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300045.png" /> in various spaces, using the metric of these spaces, by rational functions, an important role is played by characteristics of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300046.png" /> analogous to the [[Analytic capacity|analytic capacity]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300047.png" />. In terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300048.png" /> the problem of the description of all compact sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300049.png" /> on which any continuous function can be approximated with arbitrary accuracy by rational functions is solved as follows: It is necessary and sufficient that either

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300050.png" /></td> </tr></table>

for any disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300052.png" />; or that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300053.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300054.png" /> (the equivalence of a) and b) expresses the so-called  "instability"  of the analytic capacity).

3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300055.png" /> is bounded and Lebesgue-measurable and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300056.png" />, then the set of all rational functions is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300057.png" />.

4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300058.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300059.png" /> is a simply-connected domain with a rectifiable Jordan boundary, then the family of all polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300060.png" /> is dense in the [[Smirnov class|Smirnov class]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300061.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300062.png" /> is a [[Smirnov domain|Smirnov domain]].

5) Let the complex-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300064.png" />, be continuous on a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300065.png" />. Among all generalized polynomials

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300066.png" /></td> </tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300067.png" /> are arbitrary complex numbers) a generalized polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300068.png" /> deviates least from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300069.png" /> in the metric

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300070.png" /></td> </tr></table>

if and only if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300071.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300072.png" /></td> </tr></table>

for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300073.png" />.

6) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300074.png" /> is a compact set with connected complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300075.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300076.png" /> has a [[Green function|Green function]] (for the first boundary value problem for the Laplace equation) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300077.png" /> with a pole at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300078.png" />, then for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300079.png" /> and each polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300080.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300081.png" />, the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300082.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300083.png" /> holds.

7) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300084.png" /> is a bounded non-degenerate continuum with connected complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300085.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300086.png" /> is analytic at interior points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300087.png" /> and continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300088.png" /> with modulus of continuity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300089.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300090.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300091.png" /></td> </tr></table>

If the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300092.png" /> is bounded by an analytic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300093.png" />, then the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300094.png" /></td> </tr></table>

is equivalent to the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300095.png" /> is Hölder continuous of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300097.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300098.png" />. The case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a01300099.png" /> is a piecewise-smooth curve with corners has been studied.

8) In a number of cases for the approximation of analytic functions various interpolation processes prove effective, including [[Padé approximation|Padé approximation]] and its generalizations.

9) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000100.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000101.png" /> there exist both non-closed Jordan curves on which not every continuous function can be uniformly approximated by polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000102.png" /> to any degree of accuracy, and closed Jordan curves on which polynomials uniformly approximate any continuous function. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000103.png" /> this is impossible.

10) Up till now (1983) there are comparatively few direct theorems on the approximation by rational functions with free poles (i.e. without any condition on the position of the poles of the approximating functions) and a considerable amount of inverse theorems.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.L. Goncharov,   "The theory of interpolation and approximation of functions" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Walsh,   "Interpolation and approximation by rational functions in the complex domain" , Amer. Math. Soc.  (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Smirnov,   A.N. Lebedev,   "Functions of a complex variable: constructive theory" , M.I.T.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.K. Dzyadyk,   "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.N. Rusak,   "Rational functions as approximating tool" , Minsk  (1979)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  T.W. Gamelin,   "Uniform algebras" , Prentice-Hall  (1969)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.F. Leont'ev,   "Exponential series" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> , ''Some questions in approximation theory'' , Moscow  (1963)  (In Russian; translated from English)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M.V. Keldysh,   ''Dokl. Akad. Nauk SSSR'' , '''4'''  (1936)  pp. 163–166</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M. [M.A. Lavrent'ev] Lafrientieff,   "Zur Theorie der konformen Abbildung"  ''Trudy Mat. Inst. Steklov.'' , '''5'''  (1934)  pp. 159–245</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  A.N. Kolmogorov,   "Remark on the polynomials of P.L. Chebyshev deviating least from a given function"  ''Uspekhi Mat. Nauk'' , '''3''' :  1  (1948)  pp. 216–221  (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  S.N. Mergelyan,   "Uniform approximation to functions of a complex variable"  ''Series and approximation'' , '''3''' , Amer. Math. Soc.  (1962)  pp. 294–391  ''Uspekhi Mat. Nauk'' , '''7''' :  2  (1952)  pp. 31–122</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  A.G. Vitushkin,   "The analytic capacity of sets in problems of approximation theory"  ''Russian Math. Surveys'' , '''22'''  (1967)  pp. 167–200  ''Uspekhi Mat. Nauk'' , '''22''' :  6  (1967)  pp. 141–199</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  M.M. Dzhrbasyan,   "Some questions of the theory of weighted polynomials in a complex domain"  ''Mat. Sb.'' , '''36'''  (1955)  pp. 353–440  (In Russian)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  A.A. Gonchar,   "The rate of approximation of functions by rational numbers and properties of numbers" , ''Proc. Internat. Congress Mathematicians (Moscow, 1966)'' , Mir  (1968)  pp. 329–356  (In Russian)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  E.P. Dolzhenko,   P.L. Ul'yanov,   ''Vestn. Moskov. Univ. Ser. Mat. Mekh.'' , '''1'''  (1980)  pp. 3–13</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top">  S.N. Mergelyan,   "On the approximation of functions of a complex variable" , ''Mathematics in the USSR during 40 years: 1917–1957'' , '''1''' , Moscow  (1959)  pp. 383–398  (In Russian)</TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top">  A.A. Gonchar,   S.N. Mergelyan,  , ''History of paternal mathematics'' , '''1''' , Kiev  (1970)  pp. 112–193  (In Russian)</TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top">  P.M. Tamrazov,   "Smoothness and polynomial approximation" , Kiev  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top">  M.S. Mel'nikov,   S.O. Sinanyan,  , ''Contemporary problems in mathematics'' , '''4''' , Moscow  (1975)  pp. 143–250  (In Russian)</TD></TR></table>

''E.P. Dolzhenko''

====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000104.png" /> be a compact set. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000105.png" /> denote the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000106.png" /> that are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000107.png" /> and analytic at interior points (if any) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000108.png" />. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000109.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000110.png" />) the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000111.png" /> that can be uniformly approximated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000112.png" /> by polynomials (respectively, rational functions). Mergelyan's theorem states: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000113.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000114.png" />, the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000115.png" />, is connected; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000116.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000117.png" /> is finitely-connected. Examples of compact sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000118.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000119.png" /> are known (e.g. the Schweizer Käse of A. Roth, cf. [[#References|[a1]]]). The problem arises of characterizing those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000120.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000121.png" />. The solution was given by E. Bishop and A.G. Vitushkin (independently) for compact sets without interior points, and by Vitushkin for general compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000122.png" />. Vitushkin's theorem can be found in [[#References|[a1]]].

The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000124.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000125.png" /> are uniform algebras, e.g. function algebras endowed with the sup-norm (cf. [[Algebra of functions|Algebra of functions]]; [[Uniform algebra|Uniform algebra]]). For studies stressing this aspect see [[#References|[6]]] and [[#References|[a4]]].

Another direction of research is to study not approximation by linear combinations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000126.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000127.png" /> (polynomials or rational functions) but by linear combinations of powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000128.png" />, or of exponentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000129.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000130.png" />, on sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000131.png" />, mostly on curves satisfying some  "oscillation condition" . Results of this type related to the [[Müntz theorem|Müntz theorem]]; [[Lacunary power series|Lacunary power series]]; the [[Paley–Wiener theorem|Paley–Wiener theorem]], zero sets of analytic functions, etc. (cf. [[#References|[a5]]]).

In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000132.png" /> approximation problems depend essentially on the geometry of the domain under consideration (this is related to the so-called [[Levi problem|Levi problem]] of characterizing domains of holomorphy). One way to approach approximation problems is via Hörmander's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000134.png" /> mechanism (cf. [[#References|[a6]]]). An example of an approximation theorem is Kerzman's theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000135.png" /> be a strongly pseudo-convex domain with sufficiently smooth boundary (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000136.png" /> suffices). Then every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000137.png" /> holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000138.png" /> and continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000139.png" /> can be uniformly approximated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000140.png" /> by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000141.png" /> that are holomorphic on some (strongly pseudo-convex) domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000142.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000143.png" />, [[#References|[a7]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Gaier,   "Vorlesungen über Approximation im Komplexen" , Birkhäuser  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.C. Buck,   "Survey of recent Russian literature on approximation"  R.E. Langer (ed.) , ''On numerical approximation'' , Univ. of Wisconsin Press  (1959)  pp. 341–359</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Korevaar,   "Polynomial and rational approximation in the complex domain"  J.G. Clunie (ed.) , ''Aspects of contemporary complex analysis'' , Acad. Press  (1980)  pp. 251–292</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.L. Stout,   "The theory of uniform algebras" , Bogden &amp; Quigley  (1971)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.M. Redheffer,   "Completeness of sets of complex exponentials"  ''Adv. in Math.'' , '''24'''  (1977)  pp. 1–62</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L. Hörmander,   "An introduction to complex analysis in several variables" , North-Holland  (1973)  pp. Sect. 4.5</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  N. Kerzman,   "Hölder and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000144.png" /> estimates for solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013000/a013000145.png" /> on strongly pseudo-convex domains"  ''Commun. Pure Appl. Math.'' , '''24'''  (1971)  pp. 301–380</TD></TR></table>