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PROPERTIES OF CERTAIN SUBALGEBRAS OF DALES-DAVIE ALGEBRAS

Published online by Cambridge University Press:  09 August 2007

M. ABTAHI  and
T. G. HONARY
M. ABTAHI
Affiliation:
Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University, Tehran, 15618, I.R. Iran e-mails: m.abtahi@tmu.ac.ir, honary@saba.tmu.ac.ir
T. G. HONARY
Affiliation:
Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University, Tehran, 15618, I.R. Iran e-mails: m.abtahi@tmu.ac.ir, honary@saba.tmu.ac.ir
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Abstract

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We study an interesting class of Banach function algebras of infinitely differentiable functions on perfect, compact plane sets. These algebras were introduced by H. G. Dales and A. M. Davie in 1973, called Dales-Davie algebras and denoted by D(X, M), where X is a perfect, compact plane set and M = {M n }∞ n = 0 is a sequence of positive numbers such that M 0 = 1 and (m + n)!/M m+n ≤ (m!/M m )(n!/M n ) for m, n ∈ N. Let d = lim sup(n!/Mn )1/n and Xd = {z ∈ C : dist(z, X) ≤ d}. We show that, under certain conditions on X, every f ∈ D(X, M) has an analytic extension to X d . Let DP [D R ]) be the subalgebra of all f ∈ D(X, M) that can be approximated by the restriction to X of polynomials [rational functions with poles off X]. We show that the maximal ideal space of D P is , the polynomial convex hull of X d , and the maximal ideal space of D R is X d . Using some formulae from combinatorial analysis, we find the maximal ideal space of certain subalgebras of Dales-Davie algebras.

Keywords

Primary 46J1046J15Secondary 46J20
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 2 , May 2007 , pp. 225 - 233
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

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