It is defined that:
\begin{align} O(\omega)&=\frac{1}{\sqrt{2\pi}}\int O(t)e^{-i\omega t} \mathrm{d}t \tag{1} \\ O^{\dagger}(\omega)&=\frac{1}{\sqrt{2\pi}}\int O^{\dagger}(t)e^{-i\omega t} \mathrm{d}t \tag{2} \end{align} If I directly take the hermitian conjugate of equation (1) then I get: \begin{equation} O^{\dagger}(\omega)=\frac{1}{\sqrt{2\pi}}\int O^{\dagger}(t)e^{i\omega t} \mathrm{d}t \end{equation} This result is different from equation (2). Why?