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

$⟨f,\varphi ⟩=\underset{0}{\overset{\mathrm{\infty }}{\int }}\varphi (x)\log (x)\text{d}x$

By definition

$⟨f^{\prime },\varphi ⟩=-⟨f,{\varphi }^{\prime }⟩=-\underset{0}{\overset{\mathrm{\infty }}{\int }}{\varphi }^{\prime }(x)\log (x)\text{d}x$

Since the right hand side is a convergent integral, we can write

$⟨f^{\prime },\varphi ⟩=-\underset{ϵ\to 0}{\lim }\underset{ϵ}{\overset{\mathrm{\infty }}{\int }}{\varphi }^{\prime }(x)\log x\text{d}x$

Integrating by parts,

$⟨f^{\prime },\varphi ⟩=-\underset{ϵ\to 0}{\lim }[\underset{ϵ}{\overset{\mathrm{\infty }}{\int }}\frac{\varphi (x)}{x}\text{d}x+\varphi (ϵ)\log (ϵ)]$

Now, as $ϵ\to 0$ we have $\varphi (ϵ)\approx \varphi (0)$ and therefore

$\text{(1)}⟨f^{\prime },\varphi ⟩\approx -\underset{ϵ\to 0}{\lim }[\underset{ϵ}{\overset{\mathrm{\infty }}{\int }}\frac{\varphi (x)}{x}\text{d}x+\varphi (0)\log (ϵ)]$

The right hand side of (1) gives a meaning to (i.e., regularizes) the divergent integral

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\varphi (x)}{x}\text{d}x.$

We write

$⟨f^{\prime },\varphi ⟩=⟨\text{pf}\left[\frac{H(x)}{x}\right],\varphi ⟩$

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

$f^{\prime }(x)=\text{pf}\left[\frac{H(x)}{x}\right].$

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

$f^{\prime }(x)=\text{pf}\left[\frac{H(x)}{x}\right]+\text{pf}\left[\frac{H(-x)}{x}\right]=\text{pf}\left[\frac{1}{x}\right]$

where the pseudofunction $1/x$ is defined as the distribution

$⟨\text{pf}\left[\frac{1}{x}\right],\varphi ⟩=\underset{ϵ\to 0}{\lim }[\underset{ϵ}{\overset{\mathrm{\infty }}{\int }}\frac{\varphi (x)}{x}\text{d}x+\underset{-\mathrm{\infty }}{\overset{ϵ}{\int }}\frac{\varphi (x)}{x}\text{d}x]$

The individual terms diverge at $ϵ\to 0$ but the sum does not.

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

$\underset{\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\varphi (x)}{x}dx.$

This value is more commonly known as the Cauchy Principal Value. Template:Lectures