Question 12.14: Prove the following equalities for every f ∈ D′(R) : a) δa *......
Prove the following equalities for every f \in \mathcal {D} ^{\prime}( \mathbf{R}):
a) \delta_a * f=f \quad(a \in \mathbf{R} ) b) D^m \delta * f=D^{(m)} f, \quad m \in \mathbf{N}
Remark 12.14.1 One often meets the equality in a) in the (formal) form
\int_{-\infty}^{+\infty} f(y) \delta(x-y) d y=f(x) \quad(x \in \mathbf{R} )
a) Let \varphi \in \mathcal {D} ( \mathbf{R}). Then it holds
\langle\delta * f, \varphi\rangle=\langle f(x),\langle\delta(\tau), \varphi(x+\tau)\rangle\rangle=\langle f, \varphi\rangle .
b) For \varphi \in \mathcal {D} ( \mathbf{R}) and m ∈ N it holds
\left\langle D^m \delta * f, \varphi\right\rangle=\left\langle f(x),\left\langle D^m \delta(\tau), \varphi(x+\tau)\right\rangle\right\rangle=(-1)^m\left\langle f(x), \varphi^{(m)}(x)\right\rangle=\left\langle D^m f, \varphi\right\rangle