Chapter 21

Let $$ T(\varphi)=\int_{-\infty}^{\infty} \frac{1}{x} \varphi(x) d x $$ for every \(\varphi\) in the test space \(\boldsymbol{K} .\) Prove that \(T(\varphi)\) is a generalized function if the integral is understood in the sense of the Cauchy principal value. Hint. If \(\varphi\) vanishes outside the interval \([\mathrm{a}, \mathrm{b}]\), write $$ \int_{-\infty}^{\infty} \frac{1}{x} \varphi(x) d x=\int_{a}^{b} \frac{\varphi(x)-\varphi(0)}{x} d x+\int_{a}^{b} \frac{\varphi(0)}{x} d x $$

Let \(\mathbf{f}\) be a piecewise continuous function on \((-\infty, \infty)\), differentiable everywhere except at the points \(\mathrm{x}, x_{2}, \ldots, \mathrm{x}, \ldots\), where it has jumps $$ f\left(x_{n}+0\right)-f\left(x_{n}-0\right)=h_{n} \quad(n=1,2, \ldots) $$ Prove that the generalized derivative off (i.e., the derivative off regarded as a generalized function) is the sum of its ordinary derivative (at the points where it exists) and the generalized function $$ g(x)=\sum_{n=1}^{m} h_{n} \delta\left(x-x_{n}\right) $$ Comment. Note that \((g, \varphi)\) reduces to a finitesum for every test function \(\varphi .\)

Find the generalized derivative of the function of period \(2 \pi\) equal to $$ f(x)= \begin{cases}\frac{\pi-x}{2} & \text { if } \quad 0

Prove that the test space \(\boldsymbol{K}\) of all infinitely differentiable finite functions has "sufficiently many" functions in the sense that, given any two distinct continuous functions \(f_{1}\) and \(f_{2}\), there exists a function \(\varphi \in \mathrm{K}\) such that $$ \int_{-\infty}^{\infty} f_{1}(x) \varphi(x) d x \neq \int_{-\infty}^{\infty} f_{2}(x) \varphi(x) d x $$ Hint. Since \(\mathrm{f}(\mathrm{x})=f_{1}(x)-f_{2}(x) \neq 0\), there is a point \(x_{0}\) such that \(\mathrm{f}\left(x_{0}\right) \\# 0\), and hence an interval \([a, \beta]\) in whichf \((\mathrm{x})\) does not change sign. Let $$ \varphi(x)= \begin{cases}e^{-1 /(x-\alpha)^{2}} e^{-1 /(x-\beta)^{2}} & \text { if } \alpha