2
$\begingroup$

So, I am trying to find a reference for a proof of Schwartz Kernel Theorem for $2\pi$-Periodic functions, i.e, given a continuous and linear operator $A:\mathcal{D}(\mathbb{T}^N)\to \mathcal{D}'(\mathbb{T}^N)$ there exists $k\in\mathcal{D}'(\mathbb{T}^N\times \mathbb{T}^N)$ a unique periodic distribution such that $$\langle{A\varphi,\psi}\rangle=\langle k,\psi\otimes \varphi\rangle, \forall \varphi,\psi\in \mathcal{D}(\mathbb{T}^N).$$

I am fine with it being a Corollary of Schwartz Kernel Theorem in $\mathbb{R}^N$ (it is obvious?). I know there is a proof for smooth manifolds, but maybe for $\mathbb{T}^N$ there is simpler one?

$\endgroup$
1
$\begingroup$

Don't listen to algebraists and use a basis!

Here the Fourier basis works. For temperate distributions in $R^d$ which is in fact a bit more difficult, you can use the basis of Hermite functions (or even better the Meyer wavelet basis).

Then deduce the wanted kernel theorem from that for matrices.

Let $\mathscr{s}(\mathbb{N}^d)$ denote the space of multi-sequences indexed by multiindices $x=(x_{\alpha})_{\alpha\in\mathbb{N}^d}$ with faster than polynomial decay. Let $\mathscr{s}'(\mathbb{N}^d)$ that of multi-sequences $y=(y_{\alpha})_{\alpha\in\mathbb{N}^d}$ of at most polynomial growth. The needed kernel theorem is a bijective correspondence between continuous linear maps $L: \mathscr{s}(\mathbb{N}^d)\rightarrow \mathscr{s}'(\mathbb{N}^d)$ and matrices $A=(A_{\alpha,\beta})_{\alpha,\beta\in\mathbb{N}^d}\in \mathscr{s}'(\mathbb{N}^{2d})$. It's easy to prove by hand.

$\endgroup$
  • $\begingroup$ In what sense is the Meyer wavelet better or an alternative to the decomposition of tempered distributions in Hermite functions $f = \sum_{n \ge 0} <f,h_n>\frac{h_n}{\|h_n\|^2}$ ? $\endgroup$ – reuns Jun 13 at 17:37
  • $\begingroup$ For the Kernel Theorem it doesn't matter as long as you have some Schauder basis. BTW Meyer gives a basis not of $S$ itself but of the subspace of test functions of vanishing moments of all orders. For $S'$ you need to mod out by polynomials. I said Meyer wavelets is better because you can do time-frequency analysis with them, whereas Hermite functions are not so useful for that. If your function is say Holder of exponent $\alpha$ near some point, you can see that with wavelet coefficients but good luck reading that off the $\langle f,h_n\rangle$. $\endgroup$ – Abdelmalek Abdesselam Jun 13 at 17:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.