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くじらのお昼寝

【Haar関数系】

まずは,関数$H_{0,0}$を

$$ H_{0,0}(t) = 1 \qquad t \in [0,1] $$

と定める.$n \geq 1$と
$k \in \{0,\dots, 2^{n-1}-1 \}$に対して,関数$H_{n,k} \colon [0,1] \to \mathbb{R}$を以下の様に定義する.

$$ H_{n,k}(t)
=
\begin{cases}
1 & t \in \left[ \frac{2k}{2^n},\frac{2k+1}{2^n} \right[, \\
-1 & t \in \left[ \frac{2k+1}{2^n},\frac{2k+2}{2^n} \right], \\
0 & \text{otherwise}.
\end{cases} $$