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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 $a_n$?

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    $\begingroup$ 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? $\endgroup$
    – Ilya Zakharevich
    Commented Nov 11, 2019 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.) $\endgroup$
    – Ilya Zakharevich
    Commented Nov 11, 2019 at 5:17
  • $\begingroup$ Preprint of '97? $\endgroup$
    – Boris Tsygan
    Commented Nov 13, 2019 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... $\endgroup$
    – Boris Tsygan
    Commented Nov 13, 2019 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…) $\endgroup$
    – Ilya Zakharevich
    Commented Nov 22, 2019 at 10:24

2 Answers 2

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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.

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  • $\begingroup$ @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. $\endgroup$
    – Boris Tsygan
    Commented Nov 13, 2019 at 4:52
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Not sure why this is brought up after some years. But I think you should google "Fourier analysis on infinite torus" or something similar. The first result returns me a paper by Denis Fufaev.

Basically $\Bbb T=\Bbb R/\Bbb Z$ is the correct quotient because it has measure 1 (i.e. the probability measure). Consider $\Bbb T^\infty=\varinjlim_{n\to\infty}\Bbb T^n$ with product measure, which makes sense because we can build probability measure on infinite product.

Then $\hat f(m)=\lim_{n\to\infty}\int_{\Bbb T^n}f(x_1,\dots,x_n,0,\dots)e^{-2\pi i\langle x,\xi\rangle}dx_1\dots dx_n$ defines the Fourier series on $\Bbb Z^\infty$. I am not the expert on this but I guess some dimensional-free estimate is required in the further study.

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