# Oscillatory integral independent of a parameter

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?

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