We define:
| \[ F(p)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{+\infty}f(x)e^{-ipx}dx\] |
Note: The function \(f(x)\) does not have
to be periodic!
For \(\lim_{x \to \infty}\) the function \(f(x) \to 0\)
must be ”fast enough”, otherwise the integral can not be calculated, i.e. the Fourier-transform does
not exist. For such functions a Laplace-transformation often can replace the Fourier-transformation, but this will not be
discussed in this lecture.
\(F(p)\) is called the Fourier-transform of \(f(x)\).
The back-transformation is defined by
| \[f(x)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{+\infty} F(p)e^{+ipx} dp\] |
Remarks:
\(f(x)\Leftrightarrow F(p)\) unitary transformation
we write in the following: \begin{eqnarray*}F(p)&=&{\F}\left\{f(x)\right\}\\f(x)&=&{\F}^{-1}\left\{F(p)\right\} \end{eqnarray*}
FT is the generalization of Fourier-series for \(L\to\infty,n\to\infty,p_n\to p \mbox{ cont.}\) than \(p_n\to p,\sum\limits_n\to\int\ldots dp\)
Delta function: \(F(p)=1\rightarrow\,f(x)=\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}e^{ipx}dx=\delta(x)\,\rightarrow\,\) strange function \(\rightarrow\) later!
factor \(\frac{1}{\sqrt{2\pi}}\) sometimes different: you will also find:
| \[f(x)=\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}F(p)e^{ipx} dp \quad , \quad F(p)=\int\limits_{-\infty}^{+\infty}f(x)e^{-ipx}dx\] |
higher dimensions: \(f(\vec{x}),\;\vec x\in \mathbb{R}^n\): Fourier-transform \begin{eqnarray*}\vec{x}&=&\vect{x_1\\x_2\\\vdots\\x_n}=\vect{p_1\\\vdots\\p_n}\\ F(\vec{p})&=&\frac{1}{(\sqrt{2\pi})^n}\int\!\!\!\!\int\limits_{-\infty}^{+\infty}\ldots\int f(\vec{x})e^{-i\vec{p}\cdot\vec{x}}dx_1 dx_2 \ldots dx_n\\ f(\vec{x})&=&\frac{1}{(\sqrt{2\pi})^n}\int\!\!\!\!\int\limits_{-\infty}^{+\infty}\ldots\int f(\vec{p})e^{+i\vec{p}\cdot\vec{x}}dp_1 dp_2 \ldots dp_n \end{eqnarray*}
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© J. Carstensen (Math for MS)