Suppose $f:[-\pi,\pi]\to \mathbb{C}$ is a function such that $f\in L^1$, ie $f$ is Lebesgue measurable and $$\int_{-\pi}^{\pi}|f(x)|dx<+\infty.$$
By the Riemann Lebesgue Lemma we know that if we set $$c_k=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-ikx}dx,$$ for every integer $k\in\mathbb{Z}$, then $$\lim_{|k|\to+\infty}c_k=0,$$ ie for every $\epsilon>0$ there exist $N=N(\epsilon)\in\mathbb{N}$ such that for every $k\in\mathbb{Z}$ with $|k|>N$ one has $|c_k|<\epsilon$.
Now comes my (probably stupid) question.
Is there a way to conclude, as an "easy consequence" of the integral version of the Riemann Lebesgue lemma stated above, that the same holds even if $k$ is not restricted to integral values? (ie $k \in \mathbb{R}$).
Thanks a lot in advance.