Riemann–Lebesgue lemma

The Riemann–Lebesgue lemma states that the Fourier transform of an integrable function tends to zero at infinity. On the circle, the Fourier coefficients of an integrable function tend to zero.

Fourier transform: An integral transform that represents a function in terms of its frequency components. The lemma describes the Fourier transform’s behavior at large frequencies.
Lebesgue integral: An integral defined using measure and approximation by simple functions. The lemma applies to functions integrable in the Lebesgue sense.
Fourier series: A representation of a periodic function as a sum of complex exponentials or trigonometric functions. The lemma constrains coefficients but does not ensure that the series converges to the function.
Carleson's theorem: A theorem that Fourier series of square-integrable functions converge almost everywhere. Coefficient decay alone does not yield the almost-everywhere convergence established here.
Simple function: A measurable function taking only finitely many values, used to approximate integrable functions. Integrable functions can be approximated in norm by simple functions to prove the lemma.
L1 space: The space of measurable functions whose absolute values have finite integral, modulo equality almost everywhere. Its norm defines the integrability condition in the theorem.
Fourier inversion theorem: A theorem recovering a function from its Fourier transform under suitable hypotheses. Decay of the transform supports inversion arguments but is weaker than an inversion result.
Riesz–Fischer theorem: A theorem identifying square-summable coefficient sequences with Fourier coefficients of L2 functions. It concerns coefficient realizability and square summability rather than decay for every L1 function.
Step function: A function that is constant on each interval of a finite partition. Its Fourier transform can be computed directly and vanishes at high frequencies.
Fourier coefficient: A coefficient measuring a function’s component at a particular integer frequency on the circle. The circle version asserts that these coefficients vanish as frequency grows.

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