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ON THE FOURIER SERIES OF UNBOUNDED HARMONIC FUNCTIONS

Published online by Cambridge University Press:  01 April 2000

WOLFGANG LUSKY
Affiliation:
Fachbereich 17, Universität-Gesamthochschule, Warburger Strasse 100, D-33098 Paderborn, Germany; [email protected]
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Abstract

The Fourier series of the elements in the generalized Bergman spaces bp, q of harmonic functions over D and over [Copf ] (as well as those of holomorphic functions) is analysed. It is shown that the trigonometric system Ω = {r[mid ]k[mid ]eikϕ}k∈ℤ is never a basis of b1, 1 and b∞, 0 for any weighted L1-norm and L-norm over D. The same result holds in the special case of Bargmann–Fock space over [Copf ] (with respect to the weighted L1-norms and L-norms) which answers a question of Garling and Wojtaszczyk. On the other hand examples are given of weighted L1-norms and L-norms over [Copf ] where Ω is indeed a basis of b1, 1 and b∞, 0. Moreover, using similar methods, a weight is constructed on D where b∞, ∞ is not isomorphic to l which shows that there are weighted spaces whose Banach space classifications differ completely from those which have been characterized so far.

Type
Research Article
Copyright
The London Mathematical Society 2000

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