# Fourier series of smooth functions in infinitely many variables

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 $$a_n$$?

• 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? – Ilya Zakharevich Nov 11 '19 at 5:08
• 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.) – Ilya Zakharevich Nov 11 '19 at 5:17
• Preprint of '97? – Boris Tsygan Nov 13 '19 at 16:23
• 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... – Boris Tsygan Nov 13 '19 at 16:31
• (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…) – Ilya Zakharevich 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.
• @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. – Boris Tsygan Nov 13 '19 at 4:52