Title: Developments from Nonharmonic Fourier Series

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.

1991 Mathematics Subject Classification: 30, 42, 46

Keywords and Phrases:

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