Riemann-Lebesgue Lemma

Sometimes also called Mercer's Theorem.

$$\lim_{n\to \infty} \int^b_a K(\lambda, z)C\sin(nz)\,dz = 0$$

for arbitrarily large $C$ and ``nice'' $K(\lambda, z)$. Gradshteyn and Ryzhik (1979) state the lemma as follows. If $f(x)$ is integrable on $[-\pi,\pi]$, then

$$\lim_{t\to\infty} \int_{-\pi}^\pi f(x)\sin(tx)\,dx\to 0$$

and

$$\lim_{t\to\infty} \int_{-\pi}^\pi f(x)\cos(tx)\,dx\to 0.$$

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1101, 1979.