Proposition: Conditional coefficient extension

Suppose

is given away from $F\times \{0\}$, with ${a_{\alpha }}^{\circ }\in {\mathcal{D}}^{\prime }(X\setminus F)$. A supported finite-normal-order extension $t$ exists if and only if every ${a_{\alpha }}^{\circ }$ has an extension $a_{\alpha }\in {\mathcal{D}}^{\prime }(X)$. The coefficient extensions are unique for a given $t$, but are not necessarily unique from $t^{\circ }$.

Proof. Choose a smooth cutoff $\chi (q)=1$ near zero. Testing against $\chi (q)q^{\beta }\varphi (x)/\beta !$ isolates $a_{\beta }$, up to the known sign, since its normal derivative at zero is ${\delta }_{\alpha \beta }\varphi (x)$. Thus a supported extension $t$ uniquely determines coefficient extensions.

Conversely, given the coefficient extensions, their finite inserted sum defines $t$ and has the required restriction. Two coefficient extensions can differ by distributions supported on $F$, producing different contact terms.