zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Application of conformal and quasiconformal mappings and their properties in approximation theory. (English) Zbl 1074.30036
Kühnau, R. (ed.), Handbook of complex analysis: geometric function theory. Volume 1. Amsterdam: North Holland (ISBN 0-444-82845-1/hbk ). 493-520 (2002).
In this survey paper, the reader is acquainted with some results in the constructive theory of functions of a complex variable obtained by the author and his collaborators during the last two decades by an application of methods and results from modern geometric function theory of quasiconformal mappings. The main topic is the study of polynomial approximation of the members of $A(K) := \{$continuous functions $K \to \bbfC$ which are analytic in the interior of $K\}$, where $K$ is a compact subset of the complex plane $\bbfC$. The starting point is the Mergelyan theorem about the rate of uniform polynomial approximation of a function $f \in A(K)$, where $K$ has connected complement: $\lim_{n \to \infty} E_n(f,K) = 0$. Here $E_n(f,K) := \inf \{\Vert f-p\Vert _K: \, p \in P_n\}$, $\Vert h\Vert _K := \sup \{\vert h(z)\vert : \, z \in K\}$ and $P_n := \{$complex polynomials of degree at most $n\}$. Two complementary kinds of results are considered, namely: A “direct theorem”, in which the properties of $f$ are in the hypothesis and the rate of convergence of polynomials to $f$ is the conclusion; and an “inverse theorem”, in which the rate of convergence to $f$ is in the hypothesis and the properties of $f$ form the conclusion. In many of the results exhibited, the modulus of continuity $\omega_{f,K}(\delta ) := \sup \{\vert f(z_1) - f(z_2)\vert : \, z_1,z_2 \in K, \, \vert z_1-z_2\vert \leq \delta\} \,\, (\delta > 0)$ plays an essential role. In fact, some of the statements consider the class $A^\omega (K) := \{f \in A(K):$ there is $c = c(f,K) > 0$ such that $\omega_{f,K}(\delta ) \leq c \omega (\delta )$ for all $\delta > 0\}$, where $\omega :(0,+\infty ) \to (0,+\infty )$ is nondecreasing and satisfies $\omega (0+) = 0$ and $\omega (\delta t) \leq ct \omega (\delta )$ $(\delta > 0, \, t > 1)$ for some $c \geq 1$. Among the diverse results exhibited, there is a description of continua on which the functions of Hölder classes admit a Dzjadyk type approximation, that is, requiring a “nonuniform scale” of approximation on the boundary of $K$. There are also two further interesting sections in the paper. One of them deals with the approximation of polyanalytic functions (= solutions of the equation $\overline{\partial}^j f = 0$ for some $j \geq 1$, where $\overline{\partial} = (1/2)(\partial /\partial x + \partial /\partial y)$) on a quasidisk (= bounded Jordan domain $G$ such that any conformal mapping of the open unit disk onto $G$ can be extended to a $K$-quasiconformal homeomorphism of the extended plane onto itself, for some $K > 1$). The other one deals with analogues of the results previously stated in the paper in the setting of harmonic functions on a continuum $K \subset R^k$; here the cases $k=2, \, k \geq 3$ are to be distinguished. For the entire collection see [Zbl 1057.30001].

30E10Approximation in the complex domain
30C62Quasiconformal mappings in the plane
31A05Harmonic, subharmonic, superharmonic functions (two-dimensional)
41A10Approximation by polynomials