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Riemann-Lebesgue Lemma

The Riemann-Lebesgue Lemma, sometimes also called Mercer's theorem, states that

lim_(n->infty)int_a^bK(lambda,z)Csin(nz)dz=0
(1)

for arbitrarily large C and "nice" K(lambda,z). Gradshteyn and Ryzhik (2000) state the lemma as follows. If f(x) is integrable on [-pi,pi], then

lim_(t->infty)int_(-pi)^pif(x)sin(tx)dx=0
(2)

and

lim_(t->infty)int_(-pi)^pif(x)cos(tx)dx=0.
(3)

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References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1101, 2000.

Referenced on Wolfram|Alpha

Riemann-Lebesgue Lemma

Cite this as:

Weisstein, Eric W. "Riemann-Lebesgue Lemma." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Riemann-LebesgueLemma.html

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