Let
$$
\Lambda = \sum_j w_j\big(\delta(\cdot - q_j) + \delta(\cdot + q_j)\big) \in \mathcal{S}'(\mathbb{R}),
$$
where $|q_j| \to \infty$ and $w_j$ grow at most polynomially, so that $\Lambda$ is a tempered distribution.
For $\lambda > 0$ define the $L^2$-normalized dilation on test functions
$$
(S_\lambda \varphi)(x) = \lambda^{-1/2} \varphi(x/\lambda),
$$
and let $\Lambda_\varepsilon := S_{1/\varepsilon} \Lambda$. Then
$$
\langle \Lambda_\varepsilon, \varphi \rangle
= \varepsilon^{1/2} \sum_j w_j\big(\varphi(\varepsilon q_j) + \varphi(-\varepsilon q_j)\big).
$$
Question.
What are necessary and sufficient conditions on $(w_j, q_j)$ such that
$$
\Lambda_\varepsilon \xrightarrow[\varepsilon \to 0]{} T
\quad \text{in } \mathcal{S}'(\mathbb{R}),
$$
where the limit $T$ is supported at $0$ (equivalently
$T = \sum_{k \ge 0} c_{2k} \delta^{(2k)}$ since $\Lambda$ is even)?
In particular, is it true that the following “no-mass-in-annuli” condition is equivalent to collapse to a delta–jet? $$ \forall\, 0 < a < b, \qquad \varepsilon^{1/2} \sum_{j :\, a \le \varepsilon |q_j| \le b} |w_j| \to 0. $$
If the limit exists, how are the coefficients $c_{2k}$ expressed in terms of the asymptotic moments of $(w_j, q_j)$?
