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UNIFORM APPROXIMATION BY POLYNOMIAL, RATIONAL AND ANALYTIC FUNCTIONS

Part of: Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  01 June 2008

T. G. HONARY  and
S. MORADI
T. G. HONARY*
Affiliation:
Faculty of Mathematical Sciences and Computer Engineering, Tarbiat Moallem University, 599 Taleghani Avenue, 15618, Tehran (email: honary@saba.tmu.ac.ir)
S. MORADI
Affiliation:
Faculty of Mathematical Sciences and Computer Engineering, Tarbiat Moallem University, 599 Taleghani Avenue, 15618, Tehran (email: S_moradi@tmu.ac.ir)
*
✉For correspondence; e-mail: honary@saba.tmu.ac.ir
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Abstract

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Let K and X be compact plane sets such that . Let P(K) be the uniform closure of polynomials on K, let R(K) be the uniform closure of rational functions on K with no poles in K and let A(K) be the space of continuous functions on K which are analytic on int(K). Define P(X,K),R(X,K) and A(X,K) to be the set of functions in C(X) whose restriction to K belongs to P(K),R(K) and A(K), respectively. Let S0(A) denote the set of peak points for the Banach function algebra A on X. Let S and T be compact subsets of X. We extend the Hartogs–Rosenthal theorem by showing that if the symmetric difference SΔT has planar measure zero, then R(X,S)=R(X,T) . Then we show that the following properties are equivalent:

  1. (i) R(X,S)=R(X,T) ;

  2. (ii) and ;

  3. (iii) R(K)=C(K) for every compact set ;

  4. (iv) for every open set U in ℂ ;

  5. (v) for every p∈X there exists an open disk Dp with centre p such that

We prove an extension of Vitushkin’s theorem by showing that the following properties are equivalent:

  1. (i) A(X,S)=R(X,T) ;

  2. (ii) for every closed disk in ℂ ;

  3. (iii) for every p∈X there exists an open disk Dp with centre p such that

Keywords

uniform algebraspolynomial and rational approximationpeak pointsplanar measureVitushkin’s theorem

MSC classification

Secondary: 46J10: Banach algebras of continuous functions, function algebras 46J15: Banach algebras of differentiable or analytic functions, $H^p$-spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 3 , June 2008 , pp. 387 - 399
Copyright
Copyright © 2008 Australian Mathematical Society

References

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