  We study the general moment problem for measures on the real line, with
polynomials replaced by more general spaces of entire functions. As a
particular case, we describe measures that are uniquely determined by a
restriction of their Fourier transform to a finite interval. We apply our
results to prove an extension of a theorem by Eremenko and Novikov on the
frequency of oscillations of measures with a spectral gap (high-pass signals)
near infinity.
