Riemann–Lebesgue lemma

In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, is of importance in harmonic analysis and asymptotic analysis.

The lemma says that the Fourier transform or Laplace transform of an L¹ function vanishes at infinity.

Statement

If ƒ is L¹ integrable on R^d, that is to say, if the Lebesgue integral of |ƒ| is finite, then the Fourier transform of ƒ satisfies

\hat{f}(z):=\int_{\mathbb{R}^d} f(x) \exp(-iz \cdot x)\,dx \rightarrow 0\text{ as } |z|\rightarrow \infty.

Proof

First suppose that $f(x)=\chi _{(a,b)}(x)$ , the characteristic function of an open interval.

Then:

$\int f(x)e^{i\xi x}dx=\int _{a}^{b}e^{i\xi x}dx={\frac {e^{i\xi b}-e^{i\xi a}}{i\xi }}\rightarrow 0$ as $|\xi |\rightarrow \infty$

By additivity of limits, the same holds for an arbitrary simple function. That is, for any function $f$ of the form:

$f=\sum _{i=1}^{N}c_{i}\chi _{(a_{i},b_{i})},~~c_{i}\in \mathbb {R} ,~~a_{i}\leq b_{i}\in \mathbb {R}$

We have that:

$\lim _{|\xi |\rightarrow \infty }\int f(x)e^{i\xi x}dx=0$

Finally, let $f\in L^{1}$ be arbitrary.

Let $\epsilon \in \mathbb {R} >0$ be fixed.

Since the simple functions are dense in $L^{1}$ , there exists a simple function $g$ such that:

$\int \left\vert f(x)-g(x)\right\vert dx<\epsilon$

By our previous argument and the definition of a limit of a complex function, there exists $N\in \mathbb {N}$ such that for all $|\xi |>N$ :

$\left\vert \int g(x)e^{i\xi x}dx\right\vert <\epsilon$

By Integral of Integrable Function is Additive:

$\int f(x)e^{i\xi x}dx=\int (f(x)-g(x))e^{i\xi x}dx+\int g(x)e^{i\xi x}dx$

By the triangle inequality for complex numbers, the [triangle inequality] for integrals, multiplicativity of the absolute value, and Euler's Formula:

$\left\vert \int f(x)e^{i\xi x}dx\right\vert \leq \int \left\vert f(x)-g(x)\right\vert dx+\left\vert \int g(x)e^{i\xi x}dx\right\vert$

For all $|\xi |>N$ , the right side is bounded by $2\epsilon$ by our previous arguments. Since $\epsilon$ was arbitrary, this establishes:

$\lim _{|\xi |\rightarrow \infty }\int f(x)e^{i\xi x}dx=0$

for all $f\in L^{1}$ .

Other versions

The Riemann–Lebesgue lemma holds in a variety of other situations.

If ƒ is L¹ integrable and supported on (0, ∞), then the Riemann–Lebesgue lemma also holds for the Laplace transform of ƒ. That is,

\int _{0}^{\infty }f(t)e^{{-tz}}\,dt\to 0

as |z| → ∞ within the half-plane Re(z) ≥ 0.

A version holds for Fourier series as well: if ƒ is an integrable function on an interval, then the Fourier coefficients of ƒ tend to 0 as n → ±∞,

{\hat {f}}_{n}\ \to \ 0.

This follows by extending ƒ by zero outside the interval, and then applying the version of the lemma on the entire real line.

A similar statement is trivial for $L 2$ functions. To see this, note that the Fourier transform takes $L 2$ to $L 2$ and such functions have $l 2$ Fourier series.
A version holds for abstract measure spaces and replaces the use of exponential function by abstract functions. Yet the proof is not complicated. Please refer to the references listed at the end of the article.
However, the lemma does not hold for arbitrary distributions. For example, the Dirac delta function distribution formally has a finite integral over the real line, but its Fourier transform is a constant (the exact value depends on the form of the transform used) and does not vanish at infinity.

Applications

The Riemann–Lebesgue lemma can be used to prove the validity of asymptotic approximations for integrals. Rigorous treatments of the method of steepest descent and the method of stationary phase, amongst others, are based on the Riemann–Lebesgue lemma.

Proof

We'll focus on the one-dimensional case, the proof in higher dimensions is similar. Suppose first that ƒ is a compactly supported smooth function. Then integration by parts yields

\left|\int f(x)e^{-izx}\,dx\right|=\left|\int {\frac {1}{iz}}f'(x)e^{-izx}\,dx\right|\leq {\frac {1}{|z|}}\int |f'(x)|\,dx\rightarrow 0{\text{ as }}z\rightarrow \pm \infty .

If ƒ is an arbitrary integrable function, it may be approximated in the L¹ norm by a compactly supported smooth function g. Pick such a g so that ||ƒ − g||_L¹ < ε. Then

\limsup_{z\rightarrow\pm\infty} |\hat{f}(z)| \leq \limsup_{z\to\pm\infty} \left|\int (f(x)-g(x))e^{-ixz} \, dx\right| + \limsup_{z\rightarrow\pm\infty} \left|\int g(x)e^{-ixz} \, dx\right| \leq \varepsilon+0=\varepsilon,

and since this holds for any ε > 0, the theorem follows.

References

Bochner S., Chandrasekharan K. (1949). Fourier Transforms. Princeton University Press.
Weisstein, Eric W. "Riemann–Lebesgue Lemma". MathWorld.
Researchgate|https://www.researchgate.net/publication/301201556_Abstract_Generalized_Riemann-Lebesgue_Lemma
https://proofwiki.org/wiki/Euler%27s_Formula

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