Riemann–Lebesgue lemma

Multiply any reasonable function by a very rapidly oscillating wave and integrate. The wave's positive and negative humps are so closely spaced that, over each tiny stretch where the function is nearly constant, the contributions almost cancel. The faster the oscillation, the more complete the cancellation. The Riemann-Lebesgue lemma turns this intuition into a theorem: high-frequency Fourier coefficients of any integrable function must decay to zero.

Precisely, if f is integrable (in L1) on an interval, then integral of f(x) e^{-i n x} dx -> 0 as |n| -> infinity, and likewise the integrals of f(x) cos(n x) and f(x) sin(n x). For the Fourier transform on the line, the statement is that f-hat is a continuous function vanishing at infinity: f-hat(xi) -> 0 as |xi| -> infinity. The proof approximates f in L1 by a smooth or step function, for which the cancellation is transparent, then controls the error by the L1 norm.

An honest caveat about its limits. The lemma says only that the coefficients tend to zero; it says nothing about how fast, and slow decay is entirely possible — coefficients can go to 0 as slowly as 1/log n. In particular, c_n -> 0 does NOT imply the Fourier series converges, nor that the coefficients are summable. The converse also fails: a sequence tending to 0 need not be the Fourier coefficients of any L1 function. So the lemma is a necessary condition for being a Fourier coefficient sequence, but far from sufficient.