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Let $h: \mathbb{R} \mapsto \mathbb{C}$ be a positive definite function, continuous at the origin. (In fact, $h$ is the Fourier transform of a finite measure). Define the oscillatory integral $$Q(t) := \int_0^\infty \frac{\text{Im} \{ h(u)^t\}}{u} du, \quad t > 0.$$ Are there non-trivial examples of $h$ that makes $Q$ independent of $t$?

Is there a known class of functions $\{ h \}$ that have this property?

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    $\begingroup$ $h(u)=e^{iu}$ qualifies, doesn't it? $\endgroup$
    – Carlo Beenakker
    Commented May 31, 2020 at 10:35
  • $\begingroup$ Yes it does. Unfortunately for my purposes, it is a trivial case, i.e it's the Fourier transform of a point mass. $\endgroup$
    – zab
    Commented May 31, 2020 at 11:26

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