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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Riemann–Lebesgue lemma, named after
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
and
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
, states that the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
or
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
of an ''L''1 function vanishes at infinity. It is of importance in
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
and
asymptotic analysis In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as beco ...
.


Statement

Let f\in L^1(\R^n) be an integrable function, i.e. f\colon\R^n \rightarrow \C is a
measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...
such that :\, f\, _ = \int_ , f(x), \mathrmx < \infty, and let \hat be the Fourier transform of f, i.e. :\hat\colon\R^n \rightarrow \C, \ \xi\mapsto \int_ f(x) \mathrm^\mathrmx. Then \hat vanishes at infinity: , \hat(\xi), \to 0 as , \xi, \to\infty . Because the Fourier transform of an integrable function is continuous, the Fourier transform \hat is a continuous function vanishing at infinity. If C_0(\R^n) denotes the vector space of continuous functions vanishing at infinity, the Riemann–Lebesgue lemma may be formulated as follows: The Fourier transformation maps L^1(\R^n) to C_0(\R^n).


Proof

We will focus on the one-dimensional case n=1, the proof in higher dimensions is similar. First, suppose that f is continuous and
compactly supported In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...
. For \xi \neq 0, the substitution \textstyle x\to x+\frac leads to :\hat(\xi) = \int_ f(x) \mathrm^\mathrmx = \int_ f\left(x+\frac\right) \mathrm^ \mathrm^ \mathrmx = -\int_ f\left(x+\frac\right) \mathrm^ \mathrmx . This gives a second formula for \hat(\xi). Taking the mean of both formulas, we arrive at the following estimate: :, \hat(\xi), \le \frac\int_ \left, f(x)-f\left(x+\frac\right)\\mathrmx. Because f is continuous, \left, f(x)-f\left(x+\tfrac\right)\ converges to 0 as , \xi, \to \infty for all x \in \R. Thus, , \hat(\xi), converges to 0 as , \xi, \to \infty due to the
dominated convergence theorem In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
. If f is an arbitrary integrable function, it may be approximated in the L^1 norm by a compactly supported continuous function. For \varepsilon > 0, pick a compactly supported continuous function g such that \, f-g\, _ \leq \varepsilon. Then : \limsup_ , \hat(\xi), \leq \limsup_ \left, \int (f(x)-g(x))\mathrm^ \, \mathrmx\ + \limsup_ \left, \int g(x)\mathrm^ \, \mathrmx\ \leq \varepsilon + 0 = \varepsilon. Because this holds for any \varepsilon > 0, it follows that , \hat(\xi), \to 0 as , \xi, \to\infty .


Other versions

The Riemann–Lebesgue lemma holds in a variety of other situations. * If f \in L^1 z, \to \infty within the half-plane \mathrm(z) \geq 0. * A version holds for
Fourier coefficients A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
\hat_k of f tend to 0 as k \to \pm \infty . This follows by extending f by zero outside the interval, and then applying the version of the Riemann–Lebesgue lemma on the entire real line. * However, the Riemann–Lebesgue 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 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.


References

* * {{DEFAULTSORT:Riemann-Lebesgue lemma Asymptotic analysis Harmonic analysis Lemmas in analysis Theorems in analysis Theorems in harmonic analysis Bernhard Riemann