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 Answer 1

1
$\begingroup$

Don't listen to algebraists and use a basis!

Here the Fourier basis works. For temperate distributions in $\mathbb{R}^d$ 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.

$\endgroup$
2
  • $\begingroup$ 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, 2019 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 $\langle f,h_n\rangle$. $\endgroup$ Jun 13, 2019 at 17:58

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .