$\bigg|\int_{0}^{\infty}f(s)e^{-ias}ds\bigg| \leq C$

Question

I would like to know for which conditions on the function $f \in C^{1}([0,\infty]) \cap L^{1}([0,\infty])$ there is a constant $C > 0 $ such that $$ \bigg|\int_{0}^{\infty}f(s)e^{-ias}ds\bigg| \leq C $$ with $a \in \mathbb{R}$

I have no idea how to do this problem.

Answer 1

From the triangle inequality for complex integrals you have $$ \bigg|\int_{0}^{\infty}f(s)e^{-ias}ds\bigg| \leq \int_{0}^{\infty}|f(s)e^{-ias}|ds=\int_{0}^{\infty}|f(s)|ds $$

So you have to require $\int_{0}^{\infty}|f(s)|ds<\infty$ but this is the definition of $f$ belonging to $L^1$, and you are done.