operator

Shift

$$\begin{align} e_{t+} \, u_{\left[ n \right]} &= \ u_{\left[ n+1 \right]}\\ \\ e_{t-} \, u_{\left[ n \right]} &= \ u_{\left[ n-1 \right]} \end{align}$$

\[ \delta_{t-} \delta_{x x} \ = \ \frac{1}{2 \ \Delta t \ \Delta x^2} \left( e_{t+}^2 - 4 e_{t+} + 6 - 4 e_{t-} + e_{t-}^2 \right) \ \approx \ \frac{d^4}{dt^4} \]

\[ \left. \begin{align} \delta_{t+} &:= \frac{1}{\Delta t}\ \left( e_{t+}-1 \right) \\ \\ \delta_{t-} &:= \frac{1}{\Delta t}\ \left( 1-e_{t-} \right) \\ \\ \delta_{t} &:= \frac{1}{2 \ \Delta t}\ \left( e_{t+}-e_{t-} \right) \end{align} \right\} \ \approx \ \frac{d}{dt} \]

\[ \left. \begin{align} \mu_{t+} &:= \frac{1}{2}\ \left( e_{t+}+1 \right) \\ \\ \mu_{t-} &:= \frac{1}{2}\ \left( 1+e_{t-} \right) \\ \\ \mu_{t} &:= \frac{1}{2}\ \left( e_{t+}+e_{t-} \right) \end{align} \right\} \ \approx \ 1 \]

Identity

\[ 1 \; u_{\left[ n \right]} = \ u_{\left[ n \right]} \]

\[ \delta_{t t} \ := \ \delta_{t+} \ \delta_{t-} \ = \ \frac{1}{\Delta t^2} \left( e_{t+} - 2 + e_{t-} \right) \ \approx \ \frac{d^2}{dt^2} \]

\[ \delta_{t t t t} \ := \ \delta_{t t} \ \delta_{t t} \ = \ \frac{1}{\Delta t^4} \left( e_{t+}^2 - 4 e_{t+} + 6 - 4 e_{t-} + e_{t-}^2 \right) \ \approx \ \frac{d^4}{dt^4} \]