Given n vectors $w_1,\dots, w_n$ in $R^d$ we consider the functional $\phi$ defined on polynomials by $\phi(p)= \partial_{w_1}\dots \partial_{w_n}p(0)$ where $\partial_{w}$ is the directional derivative toward $w$. The question is: Is there a measure $\mu_{w_1,\dots,w_n}$ such that $\phi(p)=\int_{R^d}p(z)d\mu_{w_1,\dots,w_n}(z)$.
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$\begingroup$ You would at least need a signed measure: if $d=1$, $w_1=1$, $p=(x-1)^2$, then $\phi(p)$ is negative but $p$ is non-negative. $\endgroup$– Colin McQuillanCommented May 23, 2011 at 10:57
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No: the Dirac delta distribution at a point is a measure but its derivatives are not.
