Applied linear operators and spectral methods/Differentiating distributions 2

Differentiation of distributions with severe discontinuities
Now we will consider the case of differentiation of some locally continuous integrable functions whose discontinuities are more severe than simple jumps.

Example 1
Let us first look at the distribution defined by the locally integrable function

f(x) = H(x)~\log x $$ where $$H(x)$$ is the Heaviside function.

Then

\left\langle f,\phi \right\rangle = \int_0^{\infty} \phi(x)~\log(x)~\text{d}x $$ By definition

\left\langle f',\phi \right\rangle = -\left\langle f, \phi' \right\rangle = -\int_0^{\infty} \phi'(x)~\log(x)~\text{d}x $$ Since the right hand side is a convergent integral, we can write

\left\langle f', \phi \right\rangle = - \lim_{\epsilon\rightarrow 0} \int_{\epsilon}^{\infty} \phi'(x)~\log x~\text{d}x $$ Integrating by parts,

\left\langle f', \phi \right\rangle = - \lim_{\epsilon\rightarrow 0} \left[ \int_{\epsilon}^{\infty} \cfrac{\phi(x)}{x}~\text{d}x + \phi(\epsilon)~\log(\epsilon)\right] $$ Now, as $$\epsilon \rightarrow 0$$ we have $$\phi(\epsilon) \approx \phi(0)$$ and therefore
 * $$ \text{(1)} \qquad

\left\langle f', \phi \right\rangle \approx - \lim_{\epsilon\rightarrow 0} \left[ \int_{\epsilon}^{\infty} \cfrac{\phi(x)}{x}~\text{d}x + \phi(0)~\log(\epsilon)\right] $$ The right hand side of (1) gives a meaning to (i.e., regularizes) the divergent integral

\int_{0}^{\infty} \cfrac{\phi(x)}{x}~\text{d}x ~. $$ We write

{ \left\langle f', \phi \right\rangle = \left\langle \text{pf} \left[\cfrac{H(x)}{x}\right], \phi\right\rangle } $$ where $$\text{pf}[\bullet]$$ is the  pseudofunction which is defined by the right hand side of (1).

In this sense, if

f(x) = H(x)~\log x $$ then

{ f'(x) = \text{pf}\left[\cfrac{H(x)}{x}\right] ~. } $$

Example 2
Next let the function to be differentiated be

f(x) = \log |x| $$ We can write this function as

f(x) = H(x)~\log x + H(-x)~\log(-x) ~. $$ Then

{ f'(x) = \text{pf}\left[\cfrac{H(x)}{x}\right] + \text{pf}\left[\cfrac{H(-x)}{x}\right] = \text{pf}\left[\cfrac{1}{x}\right] } $$ where the pseudofunction $$1/x$$ is defined as the distribution

{ \left\langle \text{pf} \left[\cfrac{1}{x}\right], \phi\right\rangle = \lim_{\epsilon\rightarrow 0} \left[ \int_{\epsilon}^{\infty} \cfrac{\phi(x)}{x}~\text{d}x + \int_{-\infty}^{\epsilon} \cfrac{\phi(x)}{x}~\text{d}x \right] } $$ The individual terms diverge at $$\epsilon \rightarrow 0$$ but the sum does not.

In this way we have assigned a value to the usually divergent integral

\int_{\infty}^{\infty} \cfrac{\phi(x)}{x}~dx ~. $$ This value is more commonly known as the  Cauchy Principal Value.