シフト演算子の微分

ステップ関数: $H_\lambda(t) = \begin{cases} 0 & ( 0 \leq t< \lambda ) \\ 1 & ( 0 \leq \lambda < t ) \end{cases}$
シフト演算子: $h^\lambda = s \{ H_\lambda(t) \}$
$h^\lambda = s \{ H_\lambda(t) \}=s^2 \{ h_1 (\lambda , t) \}=s^3 \{ h_2 (\lambda , t) \}$

$h_1 (\lambda, t ) = - \frac{\partial}{\partial \lambda} h_2 ( \lambda , t) = \begin{cases} 0 & ( 0 \leq t< \lambda ) \\ t-\lambda & ( 0 \leq \lambda \leq t ) \end{cases}$
$h_2 (\lambda, t ) = \begin{cases} 0 & ( 0 \leq t \leq \lambda ) \\ \frac{1}{2} (t-\lambda)^2 & ( 0 \leq \lambda < t ) \end{cases}$

シフト演算子の微分: $(h^\lambda)' = s^3 \left\{\frac{\partial}{\partial \lambda} h_2 (\lambda, t)\right\} \\ = s^3 \{ -h_1 (\lambda, t) \} = -s \cdot s^2 \{ h_1 (\lambda, t) \} \\ = -s h^\lambda$