フーリエ変換の性質
フーリエ変換の性質
フーリエ変換には次の性質があります。
\[ \mathcal{F}_{x}\left[f\left(ax\right)\right]\left(\xi\right)=\frac{1}{\left|a\right|}\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\frac{\xi}{a}\right) \]
フーリエ変換には次の性質があります。
(1)線形性
\[ \mathcal{F}_{x}\left[af\left(x\right)+bg\left(x\right)\right]\left(\xi\right)=a\mathcal{F}_{x}\left[f\left(x\right)\right]\left(k\right)+b\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right) \](2)変数の定数倍
\(a\in\mathbb{R}\)とする。\[ \mathcal{F}_{x}\left[f\left(ax\right)\right]\left(\xi\right)=\frac{1}{\left|a\right|}\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\frac{\xi}{a}\right) \]
(3)平行移動
\[ \mathcal{F}_{x}\left[f\left(x-x_{0}\right)\right]\left(\xi\right)=e^{-2\pi i\xi x_{0}}\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right) \](4)変調
\[ \mathcal{F}_{x}\left[f\left(x\right)e^{2\pi i\xi_{0}x}\right]\left(\xi\right)=\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi-\xi_{0}\right) \](5)
\[ \mathcal{F}_{x}\left[x^{n}f\left(x\right)\right]\left(\xi\right)=\left(\frac{i}{2\pi}\right)^{n}\mathcal{F}_{x}^{\left(n\right)}\left[f\left(x\right)\right]\left(\xi\right) \](6)微分
\[ \mathcal{F}_{x}\left[f'\left(x\right)\right]\left(\xi\right)=2\pi i\xi\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right) \](7)\(n\)回微分
\[ \mathcal{F}_{x}\left[f^{\left(n\right)}\left(x\right)\right]\left(\xi\right)=\left(2\pi i\xi\right)^{n}\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right) \](8)積分
\[ \mathcal{F}_{x}\left[\int_{-\infty}^{x}f\left(t\right)dt\right]\left(\xi\right)=\frac{1}{2\pi i\xi}\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right) \](9)
\[ \mathcal{F}_{x}'\left[f\left(x\right)\right]\left(\xi\right)=-2\pi i\mathcal{F}_{x}\left[xf\left(x\right)\right]\left(\xi\right) \](10)対称性
\[ \mathcal{F}_{x}^{2\circ}\left[f\left(x\right)\right]\left(\xi\right)=f\left(-\xi\right) \](11)複素共役
\[ \mathcal{F}_{x}\left[\overline{f\left(x\right)}\right]\left(\xi\right)=\overline{\mathcal{F}_{x}\left[f\left(x\right)\right]\left(-\xi\right)} \](12)畳み込み積分
\[ \mathcal{F}_{x}\left[f\left(x\right)*g\left(x\right)\right]\left(\xi\right)=\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right) \](13)積
\[ \mathcal{F}_{x}\left[f\left(x\right)g\left(x\right)\right]\left(\xi\right)=\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)*\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right) \]-
\(\left(f*g\right)\left(x\right)\)は畳み込み\[ \left(f*g\right)\left(x\right)=\int_{-\infty}^{\infty}f\left(x'\right)g\left(x-x'\right)dx' \] である。
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3つの畳み込みの場合は、畳み込みの結合性より、\begin{align*} \mathcal{F}_{x}\left[f\left(x\right)*g\left(x\right)*h\left(x\right)\right]\left(\xi\right) & =\mathcal{F}_{x}\left[f\left(x\right)*\left(g\left(x\right)*h\left(x\right)\right)\right]\left(\xi\right)\\ & =\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)\mathcal{F}_{x}\left[g\left(x\right)*h\left(x\right)\right]\left(\xi\right)\\ & =\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right)\mathcal{F}_{x}\left[h\left(x\right)\right]\left(\xi\right) \end{align*} となる。
3つの積の場合は、積の結合性より、
\begin{align*} \mathcal{F}_{x}\left[f\left(x\right)g\left(x\right)h\left(x\right)\right]\left(\xi\right) & =\mathcal{F}_{x}\left[f\left(x\right)\left(g\left(x\right)h\left(x\right)\right)\right]\left(\xi\right)\\ & =\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)*\mathcal{F}_{x}\left[g\left(x\right)h\left(x\right)\right]\left(\xi\right)\\ & =\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)*\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right)*\mathcal{F}_{x}\left[h\left(x\right)\right]\left(\xi\right) \end{align*} となる。
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フーリエ変換の定義による違い\[ \begin{array}{|c|c|c|c|} \hline f\left(x\right) & F_{1}\left(\xi\right)=\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi ix\xi}dx & F_{2}\left(k\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)e^{-ixk}dx & F_{3}\left(\nu\right)=\int_{-\infty}^{\infty}f\left(x\right)e^{-ix\nu}d\xi\\ \hline af\left(x\right)+bg\left(x\right) & aF_{1}\left(\xi\right)+bF_{1}\left(\xi\right) & aF_{2}\left(k\right)+bF_{2}\left(k\right) & aF_{3}\left(\nu\right)+bF_{3}\left(\nu\right)\\ \hline f\left(ax\right) & \frac{1}{\left|a\right|}F_{1}\left(\frac{\xi}{a}\right) & \frac{1}{\left|a\right|}F_{2}\left(\frac{k}{a}\right) & \frac{1}{\left|a\right|}F_{3}\left(\frac{\nu}{a}\right)\\ \hline f\left(x-x_{0}\right) & e^{-2\pi i\xi x_{0}}F_{1}\left(\xi\right) & e^{-ikx_{0}}F_{2}\left(k\right) & e^{-i\nu x_{0}}F\left(\nu\right)\\ \hline f\left(x\right)e^{2\pi i\xi_{0}x} & F_{1}\left(\xi-\xi_{0}\right) & F_{2}\left(k-2\pi\xi_{0}\right) & F\left(\nu-2\pi\xi_{0}\right)\\ \hline x^{n}f\left(x\right) & \left(\frac{i}{2\pi}\right)^{n}F_{1}^{\left(n\right)}\left(\xi\right) & i^{n}F_{2}^{\left(n\right)}\left(k\right) & i^{n}F^{\left(n\right)}\left(\nu\right)\\ \hline f^{\left(n\right)}\left(x\right) & \left(2\pi i\xi\right)^{n}F_{1}\left(\xi\right) & \left(ik\right)^{n}F_{2}\left(k\right) & \left(i\nu\right)^{n}F\left(\nu\right)\\ \hline \int_{-\infty}^{x}f\left(t\right)dt & \frac{1}{2\pi i\xi}F_{1}\left(\xi\right) & \frac{1}{ik}F_{2}\left(k\right) & \frac{1}{i\nu}F\left(\nu\right)\\ \hline F\left(x\right) & F_{1}\left(-\xi\right) & F_{2}\left(-k\right) & 2\pi F_{3}\left(-\nu\right)\\ \hline \overline{f\left(x\right)} & \overline{F_{1}\left(-\xi\right)} & \overline{F_{2}\left(-k\right)} & \overline{F\left(-\nu\right)}\\ \hline \left(f*g\right)\left(x\right) & F_{1}\left(\xi\right)G_{2}\left(\xi\right) & \sqrt{2\pi}F_{2}\left(k\right)G_{2}\left(k\right) & F\left(\nu\right)G\left(\nu\right)\\ \hline f\left(x\right)g\left(x\right) & \left(F_{1}*G_{2}\right)\left(\xi\right) & \frac{1}{\sqrt{2\pi}}\left(F_{2}*G_{2}\right)\left(k\right) & \frac{1}{2\pi}\left(F*G\right)\left(\nu\right) \\\hline \end{array} \] \[ \begin{array}{|c|c|c|} \hline \mathcal{F}_{1,x}\left[f\left(x\right)\right]\left(\xi\right) & \mathcal{F}_{2,x}\left[f\left(x\right)\right]\left(k\right) & \mathcal{F}_{3,x}\left[f\left(x\right)\right]\left(\nu\right)\\ =\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi ix\xi}dx & =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)e^{-ixk}dx & =\int_{-\infty}^{\infty}f\left(x\right)e^{-ix\nu}d\xi\\ \hline \mathcal{F}_{1,x}'\left[f\left(x\right)\right]\left(\xi\right)=-2\pi i\mathcal{F}_{1,x}\left[xf\left(x\right)\right]\left(\xi\right) & \mathcal{F}_{2,x}'\left[f\left(x\right)\right]\left(k\right)=-i\mathcal{F}_{2,x}\left[xf\left(x\right)\right]\left(k\right) & \mathcal{F}_{3,x}'\left[f\left(x\right)\right]\left(\nu\right)=-i\mathcal{F}_{3,x}\left[xf\left(x\right)\right]\left(\nu\right) \\\hline \end{array} \]
(1)
\begin{align*} \mathcal{F}_{x}\left[af\left(x\right)+bg\left(x\right)\right]\left(\xi\right) & =\int_{-\infty}^{\infty}\left(af\left(x\right)+bg\left(x\right)\right)e^{-2\pi i\xi x}dx\\ & =a\int f\left(x\right)e^{-2\pi i\xi x}dx+b\int_{-\infty}^{\infty}g\left(x\right)e^{-2\pi i\xi x}dx\\ & =a\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)+b\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right) \end{align*}(2)
\begin{align*} \mathcal{F}_{x}\left[f\left(ax\right)\right]\left(\xi\right) & =\int_{-\infty}^{\infty}f\left(ax\right)e^{-2\pi i\xi x}dx\\ & =\frac{1}{a}\int_{-\sgn\left(a\right)\infty}^{\sgn\left(a\right)\infty}f\left(y\right)e^{-2\pi i\xi\frac{y}{a}}dy\cmt{y=ax}\\ & =\frac{\sgn\left(a\right)}{a}\int_{-\infty}^{\infty}f\left(y\right)e^{-2\pi i\frac{\xi}{a}y}dy\\ & =\frac{1}{\left|a\right|}\mathcal{F}_{y}\left[f\left(y\right)\right]\left(\frac{\xi}{a}\right)\\ & =\frac{1}{\left|a\right|}\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\frac{\xi}{a}\right) \end{align*}(3)
\begin{align*} \mathcal{F}_{x}\left[f\left(x-x_{0}\right)\right]\left(\xi\right) & =\int_{-\infty}^{\infty}f\left(x-x_{0}\right)e^{-2\pi i\xi x}dx\\ & =\int_{-\infty}^{\infty}f\left(y\right)e^{-2\pi i\xi\left(y+x_{0}\right)}dx\cmt{y=x-x_{0}}\\ & =e^{-2\pi i\xi x_{0}}\int_{-\infty}^{\infty}f\left(y\right)e^{-2\pi i\xi y}dx\\ & =e^{-2\pi i\xi x_{0}}\mathcal{F}_{y}\left[f\left(y\right)\right]\left(\xi\right)\\ & =e^{-2\pi i\xi x_{0}}\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right) \end{align*}(4)
\begin{align*} \mathcal{F}_{x}\left[f\left(x\right)e^{2\pi i\xi_{0}x}\right]\left(\omega\right) & =\int_{-\infty}^{\infty}f\left(x\right)e^{2\pi i\xi_{0}x}e^{-2\pi i\xi x}dx\\ & =\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi i\left(\xi-\xi_{0}\right)x}dx\\ & =\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi-\xi_{0}\right) \end{align*}(5)
\begin{align*} \mathcal{F}_{x}\left[x^{n}f\left(x\right)\right]\left(\xi\right) & =\int_{-\infty}^{\infty}x^{n}f\left(x\right)e^{-2\pi i\xi x}dx\\ & =\frac{i}{2\pi}\frac{d}{d\xi}\int_{-\infty}^{\infty}x^{n-1}f\left(x\right)e^{-2\pi i\xi x}dx\\ & =\frac{i}{2\pi}\mathcal{F}_{x}^{\left(1\right)}\left[x^{n-1}f\left(x\right)\right]\left(\xi\right)\\ & =\mathcal{F}_{x}^{\left(n\right)}\left[x^{n-n}f\left(x\right)\right]\left(\xi\right)\prod_{j=0}^{n-1}\frac{\mathcal{F}_{x}^{\left(j\right)}\left[x^{n-j}f\left(x\right)\right]\left(\xi\right)}{\mathcal{F}_{x}^{\left(j+1\right)}\left[x^{n-\left(j+1\right)}f\left(x\right)\right]\left(\xi\right)}\\ & =\mathcal{F}_{x}^{\left(n\right)}\left[x^{n-n}f\left(x\right)\right]\left(\xi\right)\prod_{j=0}^{n-1}\left(\frac{i}{2\pi}\right)\\ & =\left(\frac{i}{2\pi}\right)^{n}\mathcal{F}_{x}^{\left(n\right)}\left[f\left(x\right)\right]\left(\xi\right) \end{align*}(6)
\begin{align*} \mathcal{F}_{x}\left[f'\left(x\right)\right]\left(\xi\right) & =\int_{-\infty}^{\infty}f'\left(x\right)e^{-2\pi i\xi x}dx\\ & =\left[f\left(x\right)e^{-2\pi i\xi x}\right]_{-\infty}^{\infty}-\left(-2\pi i\xi\right)\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi i\xi x}dx\\ & =2\pi i\xi\int_{-\infty}^{\infty}f\left(x\right)e^{-i\xi x}dx\\ & =2\pi i\xi\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right) \end{align*}(7)
\begin{align*} \mathcal{F}_{x}\left[f^{\left(n\right)}\left(x\right)\right]\left(\xi\right) & =2\pi i\xi\mathcal{F}_{x}\left[f^{\left(n-1\right)}\left(x\right)\right]\left(\xi\right)\\ & =\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)\prod_{j=1}^{n}\frac{\mathcal{F}_{x}\left[f^{\left(j\right)}\left(x\right)\right]\left(\xi\right)}{\mathcal{F}_{x}\left[f^{\left(j-1\right)}\left(x\right)\right]\left(\xi\right)}\\ & =\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)\prod_{j=1}^{n}2\pi i\xi\\ & =\left(2\pi i\xi\right)^{n}\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right) \end{align*}(8)
\begin{align*} \mathcal{F}_{x}\left[\int_{-\infty}^{x}f\left(t\right)dt\right]\left(\xi\right) & =\int_{-\infty}^{\infty}\int_{-\infty}^{x}f\left(t\right)dte^{-2\pi i\xi x}dx\\ & =\left[\frac{-1}{2\pi i\xi}\int_{-\infty}^{x}f\left(t\right)dte^{-2\pi i\xi x}\right]_{-\infty}^{\infty}-\frac{-1}{2\pi i\xi}\int_{-\infty}^{\infty}f\left(t\right)e^{-2\pi i\xi x}dx\\ & =\frac{1}{2\pi i\xi}\int_{-\infty}^{\infty}f\left(t\right)e^{-2\pi i\xi x}dx\\ & =\frac{1}{2\pi i\xi}\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right) \end{align*}(9)
\begin{align*} \mathcal{F}_{x}'\left[f\left(x\right)\right]\left(\xi\right) & =\frac{d}{d\xi}\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi i\xi x}dx\\ & =\int_{-\infty}^{\infty}-2\pi ixf\left(x\right)e^{-2\pi i\xi x}dx\\ & =\mathcal{F}_{x}\left[-2\pi ixf\left(x\right)\right]\left(\xi\right)\\ & =-2\pi i\mathcal{F}_{x}\left[xf\left(x\right)\right]\left(\xi\right) \end{align*}(10)
\begin{align*} \mathcal{F}_{x}^{2\circ}\left[f\left(x\right)\right]\left(x'\right) & =\mathcal{F}_{\xi}\left[\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)\right]\left(x'\right)\\ & =\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi i\xi x}dx\cdot e^{-2\pi i\xi x'}d\xi\\ & =\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f\left(-x\right)e^{-2\pi i\xi x}dx\cdot e^{2\pi i\xi x'}d\xi\cmt{x\rightarrow-x,\xi\rightarrow-\xi}\\ & =\mathcal{F}_{\xi}^{\bullet}\left[\mathcal{F}_{x}\left[f\left(-x\right)\right]\left(\xi\right)\right]\left(x'\right)\\ & =f\left(-x'\right) \end{align*}(11)
\begin{align*} \mathcal{F}_{x}\left[\overline{f\left(x\right)}\right]\left(\xi\right) & =\int_{-\infty}^{\infty}\overline{f\left(x\right)}e^{-2\pi i\xi x}dx\\ & =\overline{\int_{-\infty}^{\infty}f\left(x\right)e^{2\pi i\xi x}dx}\\ & =\overline{\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi i\left(-\xi\right)x}dx}\\ & =\overline{\mathcal{F}_{x}\left[f\left(x\right)\right]\left(-\xi\right)} \end{align*}(12)
\begin{align*} \mathcal{F}_{x}\left[f\left(x\right)*g\left(x\right)\right]\left(\xi\right) & =\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f\left(x'\right)g\left(x-x'\right)dx'\cdot e^{-2\pi i\xi x}dx\\ & =\int_{-\infty}^{\infty}f\left(x'\right)\int_{-\infty}^{\infty}g\left(x-x'\right)e^{-2\pi i\xi x}dxdx'\\ & =\int_{-\infty}^{\infty}f\left(x'\right)\mathcal{F}_{x}\left[g\left(x-x'\right)\right]\left(\xi\right)dx'\\ & =\int_{-\infty}^{\infty}f\left(x'\right)e^{-2\pi i\xi x'}dx'\cdot\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right)\\ & =\int_{-\infty}^{\infty}f\left(x'\right)e^{-2\pi i\xi x'}dx'\cdot\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right)\\ & =\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right) \end{align*}(13)
\begin{align*} \mathcal{F}_{x}\left[f\left(x\right)g\left(x\right)\right]\left(\xi\right) & =\int_{-\infty}^{\infty}f\left(x\right)g\left(x\right)e^{-2\pi i\xi x}dx\\ & =\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi'\right)e^{2\pi i\xi'x}d\xi'g\left(x\right)e^{-2\pi i\xi x}dx\\ & =\int_{-\infty}^{\infty}\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi'\right)\int_{-\infty}^{\infty}g\left(x\right)e^{-2\pi i\left(\xi-\xi'\right)x}dxd\xi'\\ & =\int_{-\infty}^{\infty}\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi'\right)\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi-\xi'\right)d\xi'\\ & =\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)*\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right) \end{align*}
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フーリエ変換でのパーセバルの等式
\[
\int_{-\infty}^{\infty}\overline{f\left(x\right)}g\left(x\right)dx=\int_{-\infty}^{\infty}\overline{F\left(\xi\right)}G\left(\xi\right)d\xi
\]
フーリエ変換の定義とフーリエ逆変換
\[
\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)=\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi i\xi x}dx
\]
フーリエ変換の定義による違い
\[
\mathcal{F}_{2,x}\left[f\left(x\right)\right]\left(k\right)=\frac{1}{\sqrt{2\pi}}\mathcal{F}_{1,x}\left[f\left(x\right)\right]\left(\frac{k}{2\pi}\right)
\]