# 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 [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 $$.

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.

• 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.