Context. I am trying to undestand the theory underlying "Beppo-Levi"-like spaces defined as

$$ H = \left\{f\in {\cal S}'(\mathbb{R}^d) \;\left| \; t\times\widetilde{f} \in {\cal L}^2(\mathbb{R}^d) \right.\right\} $$

where $\widetilde{f}$ is the Fourier transform of the tempered distribution $f$, and $t(\xi)$ is a radial function, of class $C^\infty$, such that $1/|t|^2$ is integrable at infinity, and everywhere non-zero except in $\xi=0$ where one has $t(\xi)\sim C|\xi|^m$ for some order $m\geq 1$.

This is a direct generalization of the Beppo-Levi spaces which underlie thin-plate splines interpolation, as described e.g. by Duchon [1], Meinguet [2], Wahba [3], etc., and that correspond to taking $t(\xi)=|\xi|^m$ (with $m>d/2$ to ensure square-integrability at infinity). Just like the original Beppo-Levi spaces, $H$ contains all distributions $f\in{\cal S}'$ such that $t\times \widetilde{f}=0$, which turns out to be the space ${\cal P}_{m-1}$ of polynomials with degree $<m$.

Question. I am trying to prove that the elements of H are continuous functions. Using the Fourier inversion formula, one can show easily the following result : For every $f\in H$, if some distribution $T=\sum_{j=1}^n c_j \delta_{x_j}$ is such that $\langle T,p\rangle = 0$ for every $p\in {\cal P}_{m-1}$, then $$ T*f = \sum_{j=1}^n c_j \tau_{x_j}f $$ is a continuous function. Is this result sufficient to prove that $f$ itself is a function, moreover continous?

Indeed, for almost every point set $\{x_j\}_{j=1\dots n}$ with sufficiently many points, one can find appropriate weights $c_j$ such that distribution $T=\sum_j c_j\delta_{x_j}$ annihilates all polynomials in ${\cal P}_{m-1}$. So intuitively, I would say that the continuity of all such functions $T*f$ can only be achieved if $f$ itself is a function, moreover continuous. However, I fail to find a rigorous argument.

Remarks. This is the approach followed by Meinguet [2] in his study of the original Beppo-Levi space ($t(\xi)=|\xi|^m$). He proves the continuity of $f$ thanks to the above result, and a smart choice of distribution $T$. However, unless I misread his argument, he seems to assume without proof that the elements of $H$ are indeed functions.

Duchon [1] proves that the elements of Beppo-Levi spaces are continuous functions using a different argument, namely, their relation to Sobolev spaces. His argument however seems more specific to the original Beppo-Levi space, and I am not sure whether it can be extended to the generalization I am trying to study here.

[1] Duchon, Jean, Splines minimizing rotation-invariant semi-norms in Sobolev spaces, Constr. Theory Funct. several Variables, Proc. Conf. Oberwolfach 1976, Lect. Notes Math. 571, 85-100 (1977). ZBL0342.41012.

[2] Meinguet, Jean, An intrinsic approach to multivariate spline interpolation at arbitrary points, Polynomial and spline approximation, Proc. NATO adv. Study Inst., Calgary 1978, NATO adv. Study Inst. Ser., Ser. C - math. phys. Sci. 49, 163-190 (1979). ZBL0413.41007.

[3] Wahba, Grace, Spline models for observational data, CBMS-NSF Regional Conference Series in Applied Mathematics. 59. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. XII, 169 p. (1990). ZBL0813.62001.