Dirac delta function is not a function in the standard sense, it is a generalized function, a distribution. Are there distributions that cannot be constructed as countable linear combinations of Dirac deltas and their derivatives?

More precisely, I am asking about existence (and examples, if applicable) of distributions $g(x)$ that are not of the following form \begin{equation} g(x) = f(x) + \sum_{n=0}^\infty \sum_{k=0}^{N_n} c_{n,k} \delta^{(n)}(x-x_{n,k}), \end{equation} where $f(x)$ is just a function in the usual sense (including step functions), $N_n \in \mathbb N_0 \cup \{\infty\}$, $c_{n,k} \in \mathbb R$, $\delta^{(n)}(x)$ is the $n$-th derivative of the Dirac delta and $x_{n,k}\in\mathbb R$.

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    $\begingroup$ There is one more part in the Lebesgue decomposition of a measure. You have the first and the third. $\endgroup$ – user547557 Apr 3 '18 at 14:36

The distributional derivatives of the Cantor function are not of this form.

The first derivative can be identified with the so-called Cantor measure $\mu$, which assigns measure 1 to the Cantor set and measure 0 to every finite and countable set. It is singular to Lebesgue measure and to all Dirac measures.


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