Journal Article

CONVERGENCE OF MONOMIAL EXPANSIONS IN BANACH SPACES

Andreas Defant and Pablo Sevilla-Peris

in The Quarterly Journal of Mathematics

Volume 63, issue 3, pages 569-584
Published in print September 2012 | ISSN: 0033-5606
Published online January 2011 | e-ISSN: 1464-3847 | DOI: https://dx.doi.org/10.1093/qmath/haq053

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If E is a Banach sequence space, then each holomorphic function defines a formal power series ∑α cα(f) zα. The problem of when such an expansion converges absolutely and actually represents the function goes back to the very beginning of the theory of holomorphic functions on infinite-dimensional spaces. Several very deep results have been given for scalar-valued functions by Ryan, Lempert and Defant, Maestre and Prengel. We go on with this study, looking at monomial expansions of vector-valued holomorphic functions on Banach spaces. Some situations are very different from the scalar-valued case.

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Subjects: Pure Mathematics