Rectangular summation of multiple Fourier series and multi-parametric capacity

Perfekt, K.-M. (2021) Rectangular summation of multiple Fourier series and multi-parametric capacity. Potential Analysis, 55 (3). pp. 389-402. ISSN 0926-2601 doi: 10.1007/s11118-020-09861-5

Abstract/Summary

We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which diverges on this set, in the sense of Pringsheim. We also prove that the multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation and radial limits of Dirichlet space functions. As a by-product of the methods of proof, the results also hold in the vector-valued setting.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/91544
Identification Number/DOI	10.1007/s11118-020-09861-5
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Springer
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