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On Fourier Series in Convex Domains

Aksnes, Vegard
Master thesis
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URI
http://hdl.handle.net/11250/258376
Date
2007
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  • Institutt for matematiske fag [2686]
Abstract
We consider systems of complex exponential functions in spaces of square integrable functions. Some classical one-dimensional theory is reviewed, in particular, we emphasize the duality between the Riesz bases of complex exponential functions in $L^2$-spaces and complete interpolating sequences in $PW^2$-spaces of entire functions of exponential type. Basis properties for $L^2$-spaces over planar convex domains are then studied in detail. The convex domain in question is shown to be crucial for what basis properties the corresponding $L^2$-space possesses. We explain some results related to Fuglede's conjecture about existence of orthonormal bases and then a result by Lyubarskii and Rashkovskii regarding Riesz bases for $L^2$-spaces over convex polygons, symmetric with respect to the origin. Finally, we make a modest attempt to apply the techniques by Lyubarskii and Rashkovskii combined with approximation of plurisubharmonic functions using logarithms of moduli of entire functions, to construct a complete system of exponential functions in the space of square integrable functions over a disk. This work is not completed yet.
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Institutt for matematiske fag

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