Paley-Wiener theorem

The Paley-Wiener theorem , named after Raymond Paley and Norbert Wiener , is a theorem from the mathematical branch of functional analysis . He characterizes the Fourier-Laplace transformations of smooth functions with compact support or tempered distributions with compact support by means of growth conditions.

introduction

If a function can be integrated, it is well known that the Fourier transform can be used ${\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} }$

${\displaystyle {\hat {f}}(\xi ):=(2\pi )^{-n/2}\int _{\mathbb {R} ^{n}}f(x)e^{-\mathrm {i} \langle \xi ,x\rangle }\mathrm {d} x}$

form, where and is the scalar product of the vectors . This formula is also useful for complex vectors . Is called ${\displaystyle \xi \in \mathbb {R} ^{n}}$ ${\displaystyle \langle \xi ,x\rangle }$ ${\displaystyle \xi ,x\in \mathbb {R} ^{n}}$ ${\displaystyle \zeta \in \mathbb {C} ^{n}}$

${\displaystyle F_{f}\colon \mathbb {C} ^{n}\rightarrow \mathbb {C} ,\,F_{f}(\zeta ):=(2\pi )^{-n/2}\int _{\mathbb {R} ^{n}}f(x)e^{-\mathrm {i} \langle \zeta ,x\rangle }\mathrm {d} x}$

the Fourier-Laplace transform of . Through dualization, this concept formation can be extended to distributions with a compact carrier. If the distribution is tempered, it's done ${\displaystyle f}$ ${\displaystyle T}$

${\displaystyle {\hat {T}}(\xi ):=(2\pi )^{-n/2}\,T(x\mapsto e^{-\mathrm {i} \langle \xi ,x\rangle })}$

defines the Fourier transform. In addition, it has to be noted that there is a smooth function and that the distributions with compact support are exactly the continuous, linear functionals in the space of the smooth functions. The above formula can obviously also be written for and called ${\displaystyle x\mapsto e^{-\mathrm {i} \langle \xi ,x\rangle }}$ ${\displaystyle \zeta \in \mathbb {C} ^{n}}$

${\displaystyle F_{T}(\zeta ):=(2\pi )^{-n/2}\,T(x\mapsto e^{-\mathrm {i} \langle \zeta ,x\rangle })}$

again the Fourier-Laplace transform of . ${\displaystyle T}$

The Fourier-Laplace transforms are holomorphic functions and the question arises as to which holomorphic functions can occur here as Fourier-Laplace transforms. The Paley-Wiener sentence answers precisely this question. ${\displaystyle \mathbb {C} ^{n}\rightarrow \mathbb {C} }$

Paley-Wiener theorem for functions

A holomorphic function is the Fourier-Laplace transform of a smooth function with a carrier in the sphere if and only if there is a real constant for each such that ${\displaystyle F\colon \mathbb {C} ^{n}\rightarrow \mathbb {C} }$ ${\displaystyle \{x\in \mathbb {R} ^{n}|\,\|x\|\leq B\}}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle C_{N}>0}$

${\displaystyle |F(\zeta )|\leq C_{N}(1+\|\zeta \|)^{-N}e^{B\|\mathrm {Im} \zeta \|}}$

for everyone . ${\displaystyle \zeta \in \mathbb {C} ^{n}}$

Here, the real vector of the imaginary parts of the components of the vector . ${\displaystyle \mathrm {Im} \zeta }$ ${\displaystyle \zeta }$

Paley-Wiener set for distributions

A holomorphic function is the Fourier-Laplace transform of a distribution with a carrier in the sphere if and only if there are constants and such that ${\displaystyle F\colon \mathbb {C} ^{n}\rightarrow \mathbb {C} }$ ${\displaystyle \{x\in \mathbb {R} ^{n}|\,\|x\|\leq B\}}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle C>0}$

${\displaystyle |F(\zeta )|\leq C(1+\|\zeta \|)^{N}e^{B\|\mathrm {Im} \zeta \|}}$

for everyone . ${\displaystyle \zeta \in \mathbb {C} ^{n}}$

comment

The condition in the sentence for functions is more restrictive than the condition in the sentence for distributions. This is not surprising, because every smooth function with a compact carrier is defined by means of a distribution with a compact carrier, which is located in the carrier of , and applies to the Fourier-Laplace transforms ${\displaystyle f}$ ${\displaystyle \textstyle T_{f}(g):=\int _{\mathbb {R} ^{n}}f(x)g(x)\mathrm {d} x}$ ${\displaystyle T_{f}}$ ${\displaystyle f}$

${\displaystyle F_{T_{f}}(\zeta )=(2\pi )^{-n/2}\,T_{f}(x\mapsto e^{-\mathrm {i} \langle \zeta ,x\rangle })=(2\pi )^{-n/2}\int _{\mathbb {R} ^{n}}f(x)e^{-\mathrm {i} \langle \zeta ,x\rangle }\mathrm {d} x=F_{f}(\zeta )}$,

that is, the Fourier-Laplace transform of a smooth function with compact support is also the Fourier-Laplace transform of the distribution defined by it with compact support.

Examples

The sentences of Paley-Wiener will be explained with the help of two examples.

Be first . The Fourier-Laplace transform is ${\displaystyle f(x):=e^{-x^{2}}}$

${\displaystyle F_{f}(\zeta )={\frac {1}{\sqrt {2}}}e^{-{\frac {1}{4}}\zeta ^{2}}}$.

If the decomposition into real and imaginary parts is, that is , it grows for a fixed real part like , at least faster than for any constant . According to the sentences above, this reflects the fact that has no compact carrier, ${\displaystyle \zeta =\xi +\mathrm {i} \eta }$ ${\displaystyle \zeta ^{2}=\xi ^{2}+2\mathrm {i} \eta \xi -\eta ^{2}}$ ${\displaystyle F_{f}}$ ${\displaystyle e^{{\frac {1}{4}}\eta ^{2}}}$ ${\displaystyle e^{B|\eta |}=e^{B|\mathrm {Im} \zeta |}}$ ${\displaystyle B>0}$ ${\displaystyle f}$

Now be the distribution . A short calculation shows ${\displaystyle T}$ ${\displaystyle \textstyle T(\varphi ):=\int _{[-1,1]}\varphi (x)\mathrm {d} x}$

${\displaystyle F_{T}(\zeta )=(2\pi )^{-1/2}\int _{[-1,1]}e^{-\mathrm {i} x\zeta }\mathrm {d} x=\ldots ={\sqrt {\frac {2}{\pi }}}{\frac {\sin(\zeta )}{\zeta }}}$,

where for steadily to be continued. If the division into real and imaginary parts is true , that is , it can be estimated against , because the hyperbolic functions allow such an estimation. It follows that the growth condition from Paley-Wiener theorem for distributions is also fulfilled. Indeed it is a distribution with the compact carrier . The holomorphic function does not fulfill the condition from the Paley-Wiener theorem for functions, because if there were a constant as in the theorem, it followed ${\displaystyle \zeta =0}$ ${\displaystyle \textstyle {\sqrt {\frac {2}{\pi }}}}$ ${\displaystyle \zeta =\xi +\mathrm {i} \eta }$ ${\displaystyle \sin(\zeta )=\sin(\xi )\cosh(\eta )+\mathrm {i} \cos(\xi )\sinh(\eta )}$ ${\displaystyle |{\sin(\zeta )}|}$ ${\displaystyle e^{|\eta |}=e^{1\cdot |\mathrm {Im} \zeta |}}$ ${\displaystyle F_{T}}$ ${\displaystyle B=1}$ ${\displaystyle T}$ ${\displaystyle [-1,1]}$ ${\displaystyle F_{T}}$ ${\displaystyle N=2}$ ${\displaystyle C_{N}}$

${\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {|{\sin(\zeta )}|}{|\zeta |}}\leq {\frac {C_{2}}{(1+|\zeta |)^{2}}}e^{B|\mathrm {Im} \zeta |}}$.

Especially for real arguments , the exponential term is equal to 1 and it follows , and so the sine function for large real arguments would approach 0, which, as is well known, is not the case. Although the distribution comes from the characteristic function of the interval [-1,1], and this also has a compact support, it is not smooth. ${\displaystyle \zeta }$ ${\displaystyle \textstyle |{\sin(\zeta )}|\leq C_{2}{\sqrt {\frac {\pi }{2}}}{\frac {|\zeta |}{(1+|\zeta |)^{2}}}}$

Individual evidence

1. ^ SR Simanca: Pseudo-differential Operators , John Wiley & Sons Inc. 1991, ISBN 0-470-21688-3 , Theorem 1.2.10
2. ^ K. Yosida: Functional Analysis , Springer-Verlag 1974, ISBN 0-387-06812-0 , Chapter VI.4, The Paley-Wiener Theorems. The One-sided Laplace Transform