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Expansions with Poisson kernels and related topics

Published online by Cambridge University Press:  12 January 2010

Cristina Giannotti
Affiliation:
Dipartimento di Matematica e Informatica, University of Camerino, Via Madonna delle Carceri, 62032 Camerino (Macerata), Italy, Email: ([email protected])
Paolo Manselli
Affiliation:
Dipartimento di Matematica e Applicazioni per l'Architettura, Università di Firenze, Piazza Ghiberti 27, 50100 Firenze, Italy, Email: ([email protected])
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Abstract

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Let P(r, θ) be the two-dimensional Poisson kernel in the unit disc D. It is proved that there exists a special sequence {ak} of points of D which is non-tangentially dense for ∂D and such that any function on ∂D can be expanded in series of P(|ak|, (·)–arg ak) with coefficients depending continuously on f in various classes of functions. The result is used to solve a Cauchy-type problem for Δu = μ, where μ is a measure supported on {ak}.

MSC classification

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2010