zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Smooth equilibrium measures and approximation. (English) Zbl 1123.41005

Let Σ be a closed subset on the real line and w a nonnegative continuous function on Σ, such that w(x)x0 as x± if Σ is unbounded. Let A w be the set of functions f for which there exists a sequence of weighted polynomials {w n P n } n=1 converging to f uniformly on Σ. Here P n is a polynomial of degree at most n. A w is a subalgebra of C 0 (Σ)· Let Z w be the closed subset of Σ, such that fA w if and only if f is continuous on Σ and vanishes on Z w . The non-trivial approximation of f is possible only on the support S w of an extremal measure μ w that solves an associated equilibrium problem and is smooth, i.e. ΣS w Z w ·

In the paper, the following results are shown.

1. If x 0 Int(S w ) does not belong to Z w then μ w is smooth on some neighborhood (x 0 -δ,x 0 +δ) of x 0 ·

2. Suppose that μ w is smooth on (x 0 -δ,x 0 +δ)· Then x 0 Z w if one of the following conditions holds.

a) S w can be written as the union of finitely many intervals J k and the restriction of μ w to each J k is a doubling measure on J k .

b) μ w has a positive lower bound in a neighborhood (x 0 -δ 0 ,x 0 +δ 0 )·

As corollaries, the authors obtain all previous results for approximation as well as the solution of a problem of T. Bloom and M. Branker. A connection to level curves of homogeneous polynomials of two variables is also explored.

MSC:
41A10Approximation by polynomials
30C10Polynomials (one complex variable)
31A15Potentials and capacity, harmonic measure, extremal length (two-dimensional)