# Proof that elements of Beppo-Levi-like spaces are functions (and not just distributions)?

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 , Meinguet , Wahba , 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 $$.

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  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??

 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.

 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.

 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.

• Note that if $f$ is a function and $\sum c_j \tau_{t_j} f$ is continuous, you cannot conclude that $f$ is continuous. The simplest counterexample is a discontinuous periodic function $f$, with $t_1,t_2$ periods and $c_1+c_2=0$. The characteristic function of $\mathbb Q$ is actually a very ugly example.... Aug 26, 2022 at 22:36
• Indeed. However here we have a stronger result that $\sum c_j \tau_{x_j}f$ is continuous for any choice 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. Aug 27, 2022 at 8:45

After some more thought, I gave up Meinguet's approach above (based on sums of the form $$\sum_j c_j\tau_{x_j}f$$) for a more direct approach in order to prove continuity of the elements of $$H$$. Any feedback or criticism is welcome.

Note as above

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

Now introduce any smooth window function $$k(\xi)$$ such that $$0\leq|k|\leq 1$$, with $$k(\xi)=1$$ for $$|\xi|\leq 1/2$$ and $$k(\xi)=0$$ for $$|\xi|\geq 1$$. Define the new weighting function

$$t_k(\xi):= k(\xi) + (1-k(\xi))t(\xi)$$

that behaves like $$t$$ at infinity, but converges to a nonzero constant in $$\xi=0$$ with $$t_k(0)=k(0)=1$$. Assume also that $$t_k$$ is everywhere nonzero (this is always possible by modifying function $$k$$). Hence, function $$1/t_k$$ is well-defined and belongs to $${\cal L}^2$$ (given our hypotheses on $$t$$). By a classic argument, this implies that the modified function space associated to the weighting function $$t_k$$:

$$H_k := \left\{f\in {\cal S}'(\mathbb{R}^d) \;\left| \; t_k\times\widetilde{f} \in {\cal L}^2(\mathbb{R}^d) \right.\right\}$$

is a Reproducing Kernel Hilbert Space (RKHS), with elements of the form $$f\in H_k \Longleftrightarrow f=K_k*u$$ with $$K_k\in{\cal L}^2$$ the inverse Fourier transform of $$1/t_k$$, and $$u$$ any function in $${\cal L}^2$$. In particular, such $$f\in H_k$$ is always a continuous function.

Going back to the original function space $$H$$, any $$f\in H$$ can be decomposed as $$f=f_1+f_2$$ such that

$$\widetilde{f_1} = k.\widetilde{f} \quad{\rm and}\quad \widetilde{f_2}= (1-k).\widetilde{f}$$

Then, $$f_2$$ is easily shown to belong to $$H_k$$ above, and thus, is a continuous function. On the other hand, $$\widetilde{f_1}$$ is a distribution with compact support, so by a classic result its inverse Fourier transform $$f_1$$ is a $$C^\infty$$ function. Hence, the sum $$f=f_1+f_2$$ is indeed a continuous function.