Can somebody point me to examples of distributions which are not sums of delta functions or derivatives thereof. Also of course not integrals with local integrable functions.
Additional: is there a classification?
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How about $\phi\mapsto\lim_{\epsilon\to 0^+}\left(\int_{-\infty}^{-\epsilon}\frac{\phi(x)}{x},\mathrm dx+\int_{\epsilon}^{\infty}\frac{\phi(x)}{x},\mathrm dx\right)$? – Hagen von Eitzen Apr 18 '15 at 13:02
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I think the first derivative of the so-called Minkowski's Question Mark function may be one such example. – Joelafrite Apr 18 '15 at 13:05
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Joelafrite made a good suggestion: consider the first distributional derivative of a function $f$ that is not absolutely continuous. By definition, this distribution acts as $$\phi\to -\int f\phi'$$
- If $f$ is increasing (like Cantor staircase and Minkowski's ?-function), then the distribution $f'$ is a measure.
- If $f$ has bounded variation, then $f'$ is a signed measure.
- For more extreme examples, such as the Weierstrass nowhere differentiable function, $f'$ is a distribution that is hard to imagine or compare to any familiar object.
And of course, higher distributional derivatives of the same functions provide even more singular examples.
As for classification: a derivative is of order at most $k$ if its value on test function $\phi$ is bounded by $C^k$ norm of $\phi$. So, order $0$ distributions are signed measures. There are distributions of every finite order, as well as of infinite order.
