Question 12.10: Find all distributions y such that a) y' = 0; b) y^(m) = 0, ......
Find all distributions y such that
a) y’ = 0; b) y^{(m)}=0, m ∈ N.
a) Let y \in \mathcal {D} ^{\prime}( \mathbf{R}) be a solution of the given equation, i.e.,
\left\langle y^{\prime}, \varphi\right\rangle=0 \text { for every } \varphi \in \mathcal {D} ( \mathbf{R}) \text {. } (12.30)
Let χ be a fixed test function with the property
\int_{-\infty}^{+\infty} \chi(x) d x=1 .
Now every \varphi \in \mathcal {D} ( \mathbf{R}) can be written in the form
\varphi(x)=\chi(x) \cdot \int_{-\infty}^{+\infty} \varphi(x) d x+\psi^{\prime}(x) \quad(x \in \mathbf{R} )
for some test function \psi \text {, depending on } \varphi (prove that). Then we have
\langle y, \varphi\rangle=\left\langle y, \chi(x) \cdot \int_{-\infty}^{+\infty} \varphi(x) d x+\psi^{\prime}(x)\right\rangle=\int_{-\infty}^{+\infty} \varphi(x) d x \cdot\langle y, \chi\rangle+\left\langle y, \psi^{\prime}\right\rangle .
By assumption we have
0=-\left\langle y^{\prime}, \psi\right\rangle=\left\langle y, \psi^{\prime}\right\rangle
then putting C=\int_{-\infty}^{+\infty} \psi(x) d x, we obtain
\langle y, \varphi\rangle=C \cdot \int_{-\infty}^{+\infty} \varphi(x) d x=\langle C, \varphi\rangle
for every \varphi \in \mathcal {D} ^{\prime}( \mathbf{R}). Thus the sought after solution is
y = C,
where C is an arbitrary constant.
Answer. b)y(x)=C_0+C_1 x+\cdots+C_{m-1} x^{m-1}, \text { where } C_1, C_2, \ldots, C_n are arbitrary constants.