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Riemann-Lebesgue Lemma

Sometimes also called Mercer's Theorem.

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

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

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

and

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


References

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




© 1996-9 Eric W. Weisstein
1999-05-25