Let $J$ be a set (usually countable). Let $t_j$, $j\in J$, be variables in ${\mathbb R}/2\pi i{\mathbb Z}.$ Put $u_j=\exp(it_j),$ $j\in J.$ Introduce the following semi-norms on the space of Fourier polynomials ${\mathbb C}[u_j^{\pm 1}|j\in J]:$ for $N\geq 0,$ put $$||f||_N=\max_{t\in (S^1)^J} \sum _{|\alpha|=N} |\partial ^{[\alpha ]} f(t)|$$ where the sum is taken over multi-indices $\alpha=(\alpha_j|j\in J)$; $\alpha_j\geq 0;$ $|\alpha|=\sum_{j\in J}\alpha_j;$ and $\partial ^{[\alpha ]}=\prod_{j\in J} \frac{1}{\alpha_j !}(\frac{\partial}{\partial t_j})^{\alpha_j}$.

Let $C^\infty ((S^1)^J)$ be the completion of ${\mathbb C}[u_j^{\pm 1}|j\in J]$ in the topology defined by these semi-norms.

Question. How to describe $C^\infty((S^1)^J)$ in terms of the Fourier coefficients $a_n$?

  • 1
    $\begingroup$ Are you sure you want to consider these metrics? The usefulness of Cᵏ lies in a bound for k⸣th derivatives leading to a bound on variation of k-1⸣st derivatives. However (if I did not miss something) you essentially work in (a class of) metrics on a torus for which the diameter of the torus is infinite. Would not it be better to “allow more metrics on the torus” (leading to more metrics on C∞) so that the estimate mentioned above works? $\endgroup$ Nov 11 '19 at 5:08
  • $\begingroup$ Just an example: when I have been investigating the Jacobian of a curve of genus ∞, I was using the topology which is essentially equivalent to a direct product topology. This topology is “infinitely coarser” than the topology on the torus you (apparently) want to consider… (It is in my preprint of ∼’95.) $\endgroup$ Nov 11 '19 at 5:17
  • $\begingroup$ Preprint of '97? $\endgroup$ Nov 13 '19 at 16:23
  • $\begingroup$ Yes, coarser topology is good, if one can find it in such a way that the automorphisms and differential-difference operators that I need extend to the completion... $\endgroup$ Nov 13 '19 at 16:31
  • $\begingroup$ (Yes, it is ’97!) In the simplest setup, one takes a product of several circles of radii Rᵢ. If Rᵢ → 0, then the topology is the topology of the direct product. If (Rᵢ)∈ℓ₂, then an estimate of k⸣th derivatives gives a estimate for a variation of k-1⸣st derivative (as above). I would think that if (Rᵢ)∈ℓ₁, then most of the “expected” properties would hold. Still, I would try to start with considering the case when (Rᵢ) are rapidly decreasing. (Should not be hard to cover this case…) $\endgroup$ Nov 22 '19 at 10:24

This is an off-the-cuff answer so really a comment but too long for that. If you fix the dimension at $p$ then the space you get is $C^\infty(E_p)$, where $E_p$ is $n$–dimensional euclidean space. As $p$ increases, you get a natural projective spectrum of (nuclear Fréchet) spaces. I assume the space that you want is its projective limit—also an $(FN)$-space in the countable case. The required condition on the coefficients is that for each $p$ the restriction of the multi-sequence $(\alpha_j)$ to the first $p$ coordinates is rapidly decreasing in the sense of L. Schwartz.

  • $\begingroup$ @Ilya Zakharevich projective limits do not work for me, if I am not mistaken. What I need is that certain automorphisms of the infinite dimensional torus extend to the completion. Example: $t_n\mapsto t_{n+1}-t_1.$ Actually I need also some differential-difference operators, i.e. (infinite) sums of such automorphisms composed with partial derivatives, to extend to the completion. $\endgroup$ Nov 13 '19 at 4:52

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.