Algebras Generated by Holomorphic and Harmonic Functions on the Disc | Zendy

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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.

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