# Distributions supported on a single point

Let $d=1$.

(i) Show that if $\lambda$ is a distribution and $n\geq1$ is an integer, then $\lambda x^n=0$ if and only if $\lambda$ is a linear combination of $\delta:=\delta_{\{0\}}$ and its first $n-1$ derivatives $\delta', \delta'', \cdots, \delta^{(n-1)}$.

(ii) Show that a distribution $\lambda$ is supported on $\{0\}$ if and only if it is a linear combination of $\delta$ and finitely many of its derivatives.

(iii) Generalise (ii) to the case of general dimension $d$ (where of course one now uses partial derivatives instead of derivatives).

I don't need exact solutions, I just like to get the main ideas.

In part (i) I need a generalized version of Hadamard's lemma that includes higher order derivatives.

For part (ii), I am interested in the nontrivial direction, i.e. the "only if" part.

For part (iii) I am just interested to see the generalized statement.

In case you are wondering, I am going through Terrence Tao's notes on distributions on my own and the above question is exercise 22 of the notes.

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Let $f_0$ a test function equal to $1$ at a neighborhood of $0$. Use Taylor's formula and write $1=f_0+1-f_0$. –  Davide Giraudo Nov 13 '12 at 23:35