If $f$ Lebesgue integrable, then $\lim_{k\to\infty}\int_{\mathbb{R}}\cos(xk)f(x)dx=0$. Let $f$ be a Lebesgue integrable function on $\mathbb{R}$ and let $g$ be defined on $\mathbb{R}$ by $g(y)=\int_{-\infty}^{\infty} \cos(xy)f(x)dx$.
Prove that $\lim_{k\rightarrow\infty}g(k)=0$.

Your $g$ is the imaginary part of the Fourier transform of a real function $L^1(\mathbb R)$. It is one of the basic properties of the Fourier transform that it maps $L^1$ into the space of continuous function decaying for $y\to\pm\infty$, this is the Riemann-Lebesgue lemma.
