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Algebras Generated by Holomorphic and Harmonic Functions on the Disc
Author(s) -
Izzo Alexander J.
Publication year - 2005
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609305004352
Subject(s) - holomorphic function , mathematics , bounded function , boundary (topology) , classification of discontinuities , pure mathematics , harmonic function , harmonic , algebra over a field , unit (ring theory) , mathematical analysis , physics , quantum mechanics , mathematics education
Let E be a subset of the boundary of the open unit disc D , and let A be the algebra of bounded holomorphic functions on D that extend continuously to D ∪ E . It is shown that if f is a bounded harmonic function on D that extends continuously to D ∪ E and is not holomorphic, then the uniformly closed algebra A [ f ] generated by A and f contains C ( D ¯ ) . This result contains as special cases a result on the disc algebra due to Čirka and a result on H ∞ ( D due to Axler and Shields. A stronger form of the result, in which f is allowed to have discontinuities on a small subset of E , is also established. 2000 Mathematics Subject Classification 46J10, 46J15, 30H05.