Lemma from Riemann-Lebesgue

The Riemann-Lebesgue Lemma , also the Riemann-Lebesgue Theorem , is a mathematical theorem from analysis named after Bernhard Riemann and Henri Lebesgue . It says that the Fourier transforms of integrable functions vanish at infinity .

Formulation of the sentence

Be so a measurable function with ${\displaystyle f\in L^{1}(\mathbb {R} )}$ ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }$

${\displaystyle \int _{-\infty }^{\infty }|f(x)|\mathrm {d} x<\infty }$

and the Fourier transform of , so ${\displaystyle {\hat {f}}}$ ${\displaystyle f}$

${\displaystyle {\hat {f}}\colon \mathbb {R} \rightarrow \mathbb {R} ,\xi \mapsto {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-ix\xi }\mathrm {d} x}$.

Then it vanishes in infinity, that is, or more formally, that there is a real number for each , so that for all . ${\displaystyle {\hat {f}}}$ ${\displaystyle |{\hat {f}}(\xi )|{\xrightarrow {\xi \to \pm \infty }}0}$ ${\displaystyle \epsilon >0}$ ${\displaystyle R>0}$ ${\displaystyle |{\hat {f}}(\xi )|<\epsilon }$ ${\displaystyle |\xi |>R}$

Since the Fourier transforms of integrable functions are continuous, it is a continuous function that vanishes at infinity . If one denotes the vector space of the functions that vanish at infinity with , then the Riemann-Lebesgue lemma can also be formulated as follows: The Fourier transformation to is a mapping from to . ${\displaystyle {\hat {f}}}$ ${\displaystyle C_{0}(\mathbb {R} )}$ ${\displaystyle L^{1}(\mathbb {R} )}$ ${\displaystyle L^{1}(\mathbb {R} )}$ ${\displaystyle C_{0}(\mathbb {R} )}$

proof

The proof is presented here in broad outline. To simplify matters, we initially assume that is continuous. For provides the substitution ${\displaystyle f}$ ${\displaystyle \xi \not =0}$ ${\displaystyle \textstyle x\to x+{\frac {\pi }{\xi }}}$

${\displaystyle {\hat {f}}(\xi )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-ix\xi }\mathrm {d} x={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f\left(x+{\frac {\pi }{\xi }}\right)e^{-ix\xi }e^{-i\pi }\mathrm {d} x={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }-f\left(x+{\frac {\pi }{\xi }}\right)e^{-ix\xi }\mathrm {d} x}$,

and we have a second formula for . If one now forms the mean value of both formulas and takes amounts, subtracts them under the integral, which makes the exponential term 1, it follows ${\displaystyle {\hat {f}}(\xi )}$

${\displaystyle |{\hat {f}}(\xi )|\leq {\frac {1}{2}}{\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\left|f(x)-f\left(x+{\frac {\pi }{\xi }}\right)\right|\mathrm {d} x}$.

Because of the continuity of converges against for all and . Also applies ${\displaystyle f}$ ${\displaystyle f(x)-f(x+{\frac {\pi }{\xi }})}$ ${\displaystyle 0}$ ${\displaystyle x\in \mathbf {R} }$ ${\displaystyle |\xi |\to \infty }$

${\displaystyle \int _{-\infty }^{\infty }\left|f(x)-f\left(x+{\frac {\pi }{\xi }}\right)\right|\mathrm {d} x\leq 2\int _{-\infty }^{\infty }|f(x)|\mathrm {d} x}$.

According to the theorem of majorized convergence , for converges to 0. ${\displaystyle |{\hat {f}}(\xi )|}$ ${\displaystyle |\xi |\to \infty }$

The assumption of continuity of can be dropped due to a tightness argument . Indeed, the continuous functions are close in . In other words, for each and every function there is a continuous function such that . Because of the properties of the Fourier transform, it follows that then too . As shown before, for and since there is no choice, the same statement follows for . ${\displaystyle f}$ ${\displaystyle L^{1}(\mathbb {R} )}$ ${\displaystyle \epsilon >0}$ ${\displaystyle f\in L^{1}(\mathbb {R} )}$ ${\displaystyle g\in L^{1}(\mathbb {R} )}$ ${\displaystyle \|f-g\|_{L^{1}}<\epsilon }$ ${\displaystyle \sup _{\xi \in \mathbb {R} }|{\hat {f}}(\xi )-{\hat {g}}(\xi )|<\epsilon {\sqrt {2\pi }}}$ ${\displaystyle {\hat {g}}(\xi )}$ ${\displaystyle |\xi |\to \infty }$ ${\displaystyle \epsilon }$ ${\displaystyle f}$

Generalizations

Functions of several variables

The Riemann-Lebesgue lemma can be generalized to functions : ${\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} }$

Let it be an integrable function, that is ${\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} }$

${\displaystyle \int _{-\infty }^{\infty }|f(x_{1},\ldots ,x_{n})|\mathrm {d} x_{1}\ldots \mathrm {d} x_{n}<\infty }$.

Is the Fourier transform ${\displaystyle {\hat {f}}\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} }$

${\displaystyle {\hat {f}}(\xi _{1},\ldots ,\xi _{n})={\frac {1}{(2\pi )^{n/2}}}\int _{-\infty }^{\infty }f(x)e^{-ix_{1}\xi _{1}\ldots -ix_{n}\xi _{n}}\mathrm {d} x_{1}\ldots \mathrm {d} x_{n}}$,

so applies to . ${\displaystyle |{\hat {f}}(\xi _{1},\ldots ,\xi _{n})|\rightarrow 0}$ ${\displaystyle \|(\xi _{1},\ldots ,\xi _{n})\|\to \infty }$

There is some norm on the , for example the Euclidean norm . ${\displaystyle \|\cdot \|}$ ${\displaystyle \mathbb {R} ^{n}}$

Banach algebras

The set of integrable functions, i.e. the set of L ¹ functions , forms a Banach algebra with the convolution as multiplication and the 1-norm . In the harmonic analysis one shows that the Fourier transform becomes a special case of the abstract Gelfand transform . The Riemann-Lebesgue Lemma then follows from the fact that the Gelfand representation in the space of the C ₀ depicting functions and Gelfand room from having to be identified. At the same time, the Riemann-Lebesgue lemma is generalized to locally compact Abelian groups . ${\displaystyle L^{1}(\mathbb {R} ^{n})}$ ${\displaystyle \mathbb {R} ^{n}}$

Individual evidence

1. ↑ ^{a b} M. J. Lighthill: Introduction to the Theory of Fourier Analysis and Generalized Functions , BI University Pocket Books (1966), Volume 139, ISBN 3-411-00139-9 , Chapter 4: The Riemann-Lebesgue lemma
2. ↑ Richard V. Kadison , John R. Ringrose : Fundamentals of the Theory of Operator Algebras I. Elementary Theory , Academic Press, New York (1983), ISBN 0-12-393301-3 , Corollary 3.2.28 (iii)
3. ↑ Hitoshi Kumano-go: Pseudo-differential Operators , MIT Press, Cambridge, Massachusetts (1982), ISBN 0-262-11080-6 , Chapter 1, §4, Theorem 4.1
4. ^ Walter Rudin : Fourier Analysis on Groups , 1962, Chapter 1.2.3: The Fourier Transform