Seip, Kristian

Developments from nonharmonic Fourier series

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


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


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