**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 smooth radial function (i.e., a $C^\infty$ function of $|\xi|$), 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*, so that expressions like ``$f(x_j)$'' make sense.

**EDIT:** To make my problem clearer, let me summarize Meinguet's argument below.

Letting $J:={\rm dim}({\cal P}_{m-1})$, he considers a set of points $\{a_j\}_{j=1\dots J}$ that is *unisolvant* over ${\cal P}_{m-1}$, meaning that for every possible values $(\beta_j)\in\mathbb{R}^J$, there is a *unique* polynomial $p\in{\cal P}_{m-1}$ solution to the linear system $\forall j,\; p(a_j)=\beta_j$. Let $\{p_j\}$ in ${\cal P}_{m-1}$ the dual polynomials to the points $\{a_j\}$, i.e., such that $p_j(a_i)=\delta_{ij}$. He then introduces, for every $x\in\mathbb{R}^d$, the distribution

$$ T_{(x)} := \delta_x - \sum_{j=1}^J p_j(x)\delta_{a_j} $$

One verifies instantly that $\langle T_{(x)}, p \rangle=0$ for all $p\in{\cal P}_{m-1}$, so that the above property is verified : for every $f\in H$, $T_{(x)}* f$ is a continuous function.

Moreover, using the semi-Hilbert structure of space $H$ and the existence of a reproducing kernel, he argues that for any sequence $(x_n)$ such that $x_n \to x$, one has

$$ T_{(x_n)}*f \to T_{(x)}*f $$

as continuous functions, the convergence being uniform over any compact set.

From that, he concludes directly that $f(x_n)\to f(x)$, i.e., $f$ is a continuous function. This is the part of the argument that I do not understand. Isn't this argument valid only if one *presupposes* that $f$ is indeed a function, so that expressions $f(x_n)$ and $f(x)$ actually make sense??

[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.

anychoice of offsets {$x_j$} and weights {$c_j$} such that $\sum c_j p(x_j)=0$ for all polynomials $p\in{\cal P}_{m-1}$. In particular, almost every set of offsets $\{x_j\}$ with more elements that ${\rm dim}{\cal P}_{m-1}$ can be used in the property, provided they are associated to the correct weights $c_j$. So I can hardly imagine how to build a counterexample function $f$ that would not be continuous and still verify the property. $\endgroup$