矩形関数の性質
矩形関数\(\mathrm{rect}\left(x\right)\)は次の性質を満たす。
\(\sgn\left(x\right)\)は符号関数
矩形関数\(\mathrm{rect}\left(x\right)\)は次の性質を満たす。
(1)
\[ \mathrm{rect}\left(-x\right)=\mathrm{rect}\left(x\right) \](2)
\[ \mathrm{rect}\left(x\right)=H_{\frac{1}{2}}\left(x+\frac{1}{2}\right)-H_{\frac{1}{2}}\left(x-\frac{1}{2}\right) \](3)
\[ \mathrm{rect}\left(\frac{x}{a}\right)=\left|H_{\frac{1}{2}}\left(x+\frac{a}{2}\right)-H_{\frac{1}{2}}\left(x-\frac{a}{2}\right)\right| \](4)
\[ \left|H_{\frac{1}{2}}\left(a\right)-H_{\frac{1}{2}}\left(b\right)\right|=\mathrm{rect}\left(\frac{a+b}{2\left(a-b\right)}\right) \](5)
\[ \left|\sgn\left(a\right)-\sgn\left(b\right)\right|=2\mathrm{rect}\left(\frac{a+b}{2\left(a-b\right)}\right) \](6)
\[ \mathrm{rect}\left(x\right)=H_{\frac{1}{2}}\left(\frac{1}{2}-x\right)H_{\frac{1}{2}}\left(\frac{1}{2}+x\right) \](7)
\[ \mathrm{rect}\left(x\right)=\lim_{n\rightarrow\infty}\frac{1}{1+\left|2x\right|^{n}} \]-
\(H_{c}\left(x\right)\)はヘヴィサイド関数\(\sgn\left(x\right)\)は符号関数
(1)
矩形(くけい)関数は偶関数なので明らかに成り立つ。(2)
\begin{align*} H_{\frac{1}{2}}\left(x+\frac{1}{2}\right)-H_{\frac{1}{2}}\left(x-\frac{1}{2}\right) & =\begin{cases} 1 & \left|x\right|<\frac{1}{2}\\ \frac{1}{2} & \left|x\right|=\frac{1}{2}\\ 0 & \frac{1}{2}<\left|x\right| \end{cases}\\ & =\mathrm{rect}\left(x\right) \end{align*}(3)
\begin{align*} \mathrm{rect}\left(\frac{x}{a}\right) & =H_{\frac{1}{2}}\left(\frac{x}{a}+\frac{1}{2}\right)-H_{\frac{1}{2}}\left(\frac{x}{a}-\frac{1}{2}\right)\\ & =H_{\frac{1}{2}}\left(\left|a\right|\left(\frac{x}{a}+\frac{1}{2}\right)\right)-H_{\frac{1}{2}}\left(\left|a\right|\left(\frac{x}{a}-\frac{1}{2}\right)\right)\\ & =H_{\frac{1}{2}}\left(\sgn\left(a\right)x+\frac{\left|a\right|}{2}\right)-H_{\frac{1}{2}}\left(\sgn\left(a\right)x-\frac{\left|a\right|}{2}\right)\\ & =\begin{cases} H_{\frac{1}{2}}\left(-x-\frac{a}{2}\right)-H_{\frac{1}{2}}\left(-x+\frac{a}{2}\right) & a<0\\ 0 & a=0\\ H_{\frac{1}{2}}\left(x+\frac{a}{2}\right)-H_{\frac{1}{2}}\left(x-\frac{a}{2}\right) & 0<a \end{cases}\\ & =\begin{cases} H_{\frac{1}{2}}\left(x-\frac{a}{2}\right)-H_{\frac{1}{2}}\left(x+\frac{a}{2}\right) & a<0\\ 0 & a=0\\ H_{\frac{1}{2}}\left(x+\frac{a}{2}\right)-H_{\frac{1}{2}}\left(x-\frac{a}{2}\right) & 0<a \end{cases}\\ & =\left|H_{\frac{1}{2}}\left(x+\frac{a}{2}\right)-H_{\frac{1}{2}}\left(x-\frac{a}{2}\right)\right| \end{align*} 途中で\(H_{\frac{1}{2}}\left(a\right)-H_{\frac{1}{2}}\left(b\right)=H_{\frac{1}{2}}\left(-b\right)-H_{\frac{1}{2}}\left(-a\right)\)を使っている。(4)
(3)より、\begin{align*} \left|H_{\frac{1}{2}}\left(a\right)-H_{\frac{1}{2}}\left(b\right)\right| & =\left|H_{\frac{1}{2}}\left(\frac{a+b}{2}+\frac{a-b}{2}\right)-H_{\frac{1}{2}}\left(\frac{a+b}{2}-\frac{a-b}{2}\right)\right|\\ & =\mathrm{rect}\left(\frac{a+b}{2\left(a-b\right)}\right) \end{align*} となるので与式は成り立つ。
(5)
(4)より、\begin{align*} \left|\sgn\left(a\right)-\sgn\left(b\right)\right| & =\left|2H_{\frac{1}{2}}\left(a\right)-1-\left(2H_{\frac{1}{2}}\left(b\right)-1\right)\right|\\ & =2\left|H_{\frac{1}{2}}\left(a\right)-H_{\frac{1}{2}}\left(b\right)\right|\\ & =2\mathrm{rect}\left(\frac{a+b}{2\left(a-b\right)}\right) \end{align*} となるので与式は成り立つ。
(6)
\begin{align*} H_{\frac{1}{2}}\left(\frac{1}{2}-x\right)H_{\frac{1}{2}}\left(\frac{1}{2}+x\right) & =\begin{cases} 1 & \left|x\right|<\frac{1}{2}\\ \frac{1}{2} & \left|x\right|=\frac{1}{2}\\ 0 & \frac{1}{2}<\left|x\right| \end{cases}\\ & =\mathrm{rect}\left(x\right) \end{align*}(7)
\begin{align*} \lim_{n\rightarrow\infty}\frac{1}{1+\left|2x\right|^{n}} & =\begin{cases} 1 & \left|x\right|<\frac{1}{2}\\ \frac{1}{2} & \left|x\right|=1\\ 0 & \frac{1}{2}<\left|x\right| \end{cases}\\ & =\mathrm{rect}\left(x\right) \end{align*}
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\[
\pi_{J}:X_{I}\rightarrow X_{J},\left(x_{i}\right)_{i\in I}\mapsto\left(x_{j}\right)_{j\in J}
\]
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\[
f|_{A}=f\circ i_{A}
\]
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\[
\left(f_{1}\oplus f_{2}\right)\left(x\right)=\begin{cases}
f_{1}\left(x\right) & x\in X_{1}\setminus X_{2}\\
f_{2}\left(x\right) & x\in X_{2}\setminus X_{1}\\
f_{1}\mid_{X_{1}\cap X_{2}}\left(x\right)=f_{2}\mid_{X_{1}\cap X_{2}}\left(x\right) & x\in X_{1}\cap X_{2}
\end{cases}
\]
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\[
G\left(f\right)=\left\{ \left(x,f\left(x\right)\right)\in X\times Y;x\in X\right\}
\]