Riemann-Lebesgue lemma

The Riemann-Lebesgue lemma states that the Fourier transform or Laplace transform of a signal with finite action, i.e., $x\in {\mathsf{L}}_1$ is 0 at $\infty$.

The formal statement says that a finite action signal, i.e.,

$$\Vert x{\Vert }_1={\int }_{-\infty }^{+\infty }|x(t){|}^{1}dt<\infty$$

If $X$ is the Fourier transform of $x$, then $x\to 0$ as $|\omega |\to \infty$:

$$\underset{\omega \to \pm \infty }{\lim }|X(j\omega )|=0$$