Characterization of Mergelyan sets

Stray, Arne

doi:10.1090/s0002-9939-1974-0361097-5

Cited by 13 publications

(8 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is interesting that the topological condition in Theorem 1 coincides with that obtained by Stray [14] in connection with Farrell sets for holomorphic functions, even though the two proofs have little in common and the theory of harmonic approximation differs significantly from its holomorphic counterpart (see [4]). This topological characterization also arises in connection with "Mergelyan sets" for holomorphic and harmonic functions (see [14] and [5]).…”

Section: Theorem 1 Let F Be a Relatively Closed Subset Of B Then F mentioning

confidence: 61%

“…This topological characterization also arises in connection with "Mergelyan sets" for holomorphic and harmonic functions (see [14] and [5]). We note that many authors have studied (an adapted notion of) Farrell sets for various classes of holomorphic and harmonic functions; see [2], [3], [6], [7], [8], [9], [10], [11], [12], [15] and [16].…”

Section: Theorem 1 Let F Be a Relatively Closed Subset Of B Then F mentioning

confidence: 99%

“…A set F for which this implication holds will be called a Farrell set for harmonic functions. The problem of characterizing such sets was posed by Stray [17], who had previously solved the corresponding problem for holomorphic functions on the unit disc (see [14]). A complete characterization is given in this paper.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Farrell sets for harmonic functions

Gardiner¹,

Hanley²

2002

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let F denote a relatively closed subset of the unit ball B of R n . The purpose of this paper is to characterize those sets F which have the following property: any harmonic function h on B which satisfies |h| ≤ M on F (where M > 0) can be locally uniformly approximated on B by a sequence of harmonic polynomials which satisfy the same inequality on F . This answers a question posed by Stray, who had earlier solved the corresponding problem for holomorphic functions on the unit disc.

show abstract

Section: Theorem 1 Let F Be a Relatively Closed Subset Of B Then F mentioning

confidence: 61%

Section: Theorem 1 Let F Be a Relatively Closed Subset Of B Then F mentioning

confidence: 99%

See 1 more Smart Citation

Farrell sets for harmonic functions

Gardiner¹,

Hanley²

2002

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…This paper presents a complete characterization of Mergelyan pairs for harmonic functions. The corresponding problem for holomorphic functions was solved by Stray [12] in the particular case where Ω is the unit disc, and then by Brown and Shields [2, p. 79 and Theorem 3] for general plane domains.…”

Section: Resultsmentioning

confidence: 99%

Mergelyan pairs for harmonic functions

Gardiner

1998

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let Ω ⊆ R n be open and E ⊆ Ω be a bounded set which is closed relative to Ω. We characterize those pairs (Ω, E) such that, for each harmonic function h on Ω which is uniformly continuous on E, there is a sequence of harmonic polynomials which converges to h uniformly on E. As an immediate corollary we obtain a characterization of Mergelyan pairs for harmonic functions. ResultsLet Ω be an open set in Euclidean space R n (n ≥ 2) and let E ⊆ Ω be a bounded set which is closed relative to Ω. Also, let f | E denote the restriction of a function f to E. We call (Ω, E) a Mergelyan pair for harmonic functions if each harmonic function h on Ω for which h| E is uniformly continuous on E can be uniformly approximated by harmonic polynomials on K ∪ E for every compact subset K of Ω. This paper presents a complete characterization of Mergelyan pairs for harmonic functions. The corresponding problem for holomorphic functions was solved by Stray [12] in the particular case where Ω is the unit disc, and then by Brown and Shields [2, p. 79 and Theorem 3] for general plane domains.We will need some notation. If K is a compact subset of R n , then K denotes the union of K with the bounded (connected) components of R n \K. Let A denote the Alexandroff (ideal) point for Ω. If V is a connected open subset of Ω and there is a continuous function p : [0, +∞) → V such that p(t) → A as t → +∞, then we say that A is accessible from V . We define E ∼ to be the union of E with the connected components of Ω\E from which A is not accessible. Note that the definition of E ∼ involves Ω, whereas the definition of K does not. Also, the notation E means (E) . We refer to Doob [6, 1.XI] for an account of thin sets. Theorem 1. Let Ω be an open set in Rn and E ⊆ Ω be a bounded set which is closed relative to Ω. The following are equivalent :(a) each harmonic function h on Ω for which h| E is uniformly continuous on E can be uniformly approximated on E by harmonic polynomials; (b) R n \E and R n \E ∼ are thin at the same points of E. Corollary 1.Let Ω be an open set in R n and E ⊆ Ω be a bounded set which is closed relative to Ω. The following are equivalent :(a) (Ω, E) is a Mergelyan pair for harmonic functions;

show abstract

“…See [16] and [18]. Results about Farrell sets for other function spaces can for example be found in [12], [11], [17] and [20].…”

Section: S M (F ) Is Thick At Allmost All Z ∈ F ∩ T the Excep-mentioning

confidence: 99%