Non-standard examples of Distributions

Question

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?

Answer 1

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.