For a real-valued uni-variate r.v. $X$, with pdf $f(x)$ and absolute integrable cf $\varphi(t)$, we have the following transform:$$2\pi f(x)=\int_{-\infty}^{\infty}e^{-itx}\varphi(t)\,dt.$$ However, I want to understand that, for any given $a,b\in\mathbb{R}$ (with $a<b$), what does the transform $$\mathcal F'_{a,b}\circ\varphi(t)=g(x)=\int_{a}^{b}e^{-itx}\varphi(t)\,dt$$ represent, if that makes sense at all? The closest concept seems to be the Finite Fourier transform, which I am not familiar with.
More importantly, I want to know if there exist some finite constant $M$, so that $$|g(x)|=\left|\int_{a}^{b}e^{-itx}\varphi(t)\,dt\right|\leq M.$$ Obviously $M=b-a$ if we allow $M$ to vary, but a better guess seems to be $M=\|f(x)\|_\infty$, since $\varphi(t)$ is bounded and continuous, am I correct?