Divided differences and systems of nonharmonic Fourier series

Abstract:

Suppose that ${\omega _{n,0}},{\omega _{n,1}}, \ldots ,{\omega _{n,k}}$ are distinct complex numbers with $|n - {\omega _{n,j}}| \leqslant \delta$ 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 \[ \int _{ - (k + 1)\pi }^{(k + 1)\pi } {f(t){e^{ - it{\omega _{n,j}}}}} dt = {c_{n,j}}\quad {\text {for}}\;n \in {\mathbf {Z}},j = 0,1, \ldots ,k\] 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.