Seip, Kristian

Developments from nonharmonic Fourier series

Doc. Math. (Bielefeld) Extra Vol. ICM Berlin 1998, Vol. II, 713-722 (1998)

Summary

We begin this survey by showing that Paley and Wiener's unconditional basis problem for nonharmonic Fourier series can be understood as a problem about weighted norm inequalities for Hilbert operators. Then we reformulate the basis problem in a more general setting, and discuss Beurling-type density theorems for sampling and interpolation. Next, we state some multiplier theorems, of a similar nature as the famous Beurling-Malliavin theorem, and sketch their role in the subject. Finally, we discuss extensions of nonharmonic Fourier series to weighted Paley-Wiener spaces, and indicate how these spaces are explored via de Branges' Hilbert spaces of entire functions.

Mathematics Subject Classification

42C30, 46E22, 42C40, 42B15, 30D20

Keywords/Phrases

nonharmonic Fourier series, weighted norm inequalities, Beurling-type density theorems, sampling, interpolation, multiplier theorems, weighted Paley-Wiener spaces

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