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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Divided differences and systems of nonharmonic Fourier series


Author: David Ullrich
Journal: Proc. Amer. Math. Soc. 80 (1980), 47-57
MSC: Primary 42C30; Secondary 46B15
MathSciNet review: 574507
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Abstract: Suppose that $ {\omega _{n,0}},{\omega _{n,1}}, \ldots ,{\omega _{n,k}}$ are distinct complex numbers with $ \vert n - {\omega _{n,j}}\vert \leqslant \delta $ for all $ n \in {\mathbf{Z}},j = 0,1, \ldots ,k$. We show that if $ \delta > 0$ is small enough then, given complex numbers $ {c_{n,j}}(n \in {\mathbf{Z}},j = 0,1, \ldots ,k)$ there exists $ f \in {L^2}( - (k + 1)\pi ,(k + 1)\pi )$ with

$\displaystyle \int_{ - (k + 1)\pi }^{(k + 1)\pi } {f(t){e^{ - it{\omega _{n,j}}}}} dt = {c_{n,j}}\quad {\text{for}}\;n \in {\mathbf{Z}},j = 0,1, \ldots ,k$

if and only if certain ``divided differences'' involving the $ {c_{n,j}}$'s and the $ {\omega _{n,j}}$'s are square summable. This extends a classical theorem of Paley and Wiener, which is equivalent to the case $ k = 0$ above.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0574507-8
PII: S 0002-9939(1980)0574507-8
Keywords: Nonharmonic Fourier series, Riesz basis, divided difference
Article copyright: © Copyright 1980 American Mathematical Society