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Expansions with Poisson kernels and related topics

Part of: Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  12 January 2010

Cristina Giannotti  and
Paolo Manselli
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}.

Keywords

Poisson kernelsinterpolationCauchy problem

MSC classification

Secondary: 42C30: Completeness of sets of functions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 53 , Issue 1 , February 2010 , pp. 153 - 173
Copyright
Copyright © Edinburgh Mathematical Society 2010