Rings Of Analytic Functions Research Articles

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Finitely generated ideals in rings of analytic functions

Given a domain g2 of C and a function p > 0 on f2, Ap = Ap(~d) denotes the ring of all functions f ' f 2 ~ l E analytic on f2 for which there exist constants c l = c l ( f ) ~ O and c 2 = c 2 ( f ) > 0 such that [ f ( z ) l~c l exp(c2p(z)), zef2. If F1, ..., FN e Ap, and if G ~ Ap belongs to the ideal J(F1, ..., FN) of Ap generated by F1 . . . . . F N, then it is easily seen that there must exist constants cl, c 2 ~ 0 such that IG(z)[ < cl [IF(z)H exp(c2p(z)), z ~ f2, (1.1)

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Mathematische Annalen

Oct 1, 1971
James J Kelleher + 1

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A ring of analytic functions, II

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Studia Mathematica

Jan 1, 1971
R Brooks

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Ideals in Rings of Analytic Functions with Smooth Boundary Values

LetAdenote the Banach algebra of functions analytic in the open unit discDand continuous in. Iffand its firstmderivatives belong toA,then the boundary functionf(eiθ)belongs toCm(∂D). The spaceAmof all such functions is a Banach algebra with the topology induced byCm(∂D).If all the derivatives of/ belong toA,then the boundary function belongs toC∞(∂D), and the spaceA∞all such functions is a topological algebra with the topology induced byC∞(∂D). In this paper we determine the structure of the closed ideals ofA∞(Theorem 5.3).Beurling and Rudin (see e.g. [7, pp. 82-89;10]) have characterized the closed ideals ofA, and their solution suggests a possible structure for the closed ideals ofA∞.

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