Series of tempered distributions converge to dirac distribution

Question

I want to show that for a tempered distribution $u$ and a series $\psi_k$ of smooth functions such that $\psi_k(x)=1$ for $\vert x\vert\leq 2^{-k}$ and $\psi_k=0$ for $\vert x\vert\geq2^{-k+1}$ there holds $$ \psi_ku\rightarrow\sum_{\vert \alpha\vert\leq N}c_\alpha\delta^\alpha\qquad \text{in} \qquad \mathcal{S}'$$ with some real constants $c_\alpha$ and $\delta^\alpha(\phi):=D^\alpha\phi(0)$. I'm thankful for every hint.

Answer 1

This is not true in the stated generality: if the functions $\psi_k$ do something crazy in the annulus $2^{-k-1}<|x|<2^{-k}$ (e.g., if $\int \psi_k\to\infty$), the sequence $\psi_ku$ need not converge at all.

You need some assumptions to ensure the existence of distributional limit. Instead of inventing assumptions, I'll just assume that the limit (called $T$) exists, and show that it has the claimed form.

For any test function $\phi$ with support contained in $\mathbb{R}^n\setminus \{0\}$, we have $\langle \psi_k u,\phi\rangle =0$ for sufficiently large $k$. Hence $\langle T,\phi\rangle = 0$. In other words, the support of $T$ is the single point $\{0\}$. The claim now follows from the fact (pointed out by Christian Remling) that distribution with point support are linear combinations of derivatives at that point.