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Identifying $\mathbb{R}/\mathbb{Z}$ with the interval $\left[0,1\right)$, let $C_{\textrm{coun}}\left(\mathbb{R}/\mathbb{Z}\right)$ denote the set of all functions $f:\mathbb{R}/\mathbb{Z}\rightarrow\mathbb{C}$ supported on countably many points in $\left[0,1\right)$. For each $p\in\left[1,\infty\right)$, let: $$\ell^{p}\left(\mathbb{R}/\mathbb{Z}\right)=\left\{ f\in C_{\textrm{coun}}\left(\mathbb{R}/\mathbb{Z}\right):\left\Vert f\right\Vert _{\ell^{p}\left(\mathbb{R}/\mathbb{Z}\right)}=\left(\sum_{t\in\mathbb{R}/\mathbb{Z}}\left|f\left(t\right)\right|^{p}\right)^{1/p}<\infty\right\}$$ and define $\hat{f}:\mathbb{R}\rightarrow\mathbb{C}$ by:$$\hat{f}\left(x\right)=\sum_{t\in\mathbb{R}/\mathbb{Z}}f\left(t\right)e^{2\pi itx}$$ When $f\in\ell^{1}\left(\mathbb{R}/\mathbb{Z}\right)$, it is clear that $\hat{f}$ is uniformly continuous on $\mathbb{R}$. Moreover, direct computation then shows that the inversion formula: $$f\left(t\right)=\lim_{X\rightarrow\infty}\frac{1}{2X}\int_{-X}^{X}\hat{f}\left(x\right)e^{-2\pi itx}dx$$ holds for all $t\in\mathbb{R}/\mathbb{Z}$. Does this version of the Fourier transform satisfy the usual properties? Namely:

I. If $f,g\in\ell^{2}\left(\mathbb{R}/\mathbb{Z}\right)$, then:

i. $\hat{f}$ converges almost everywhere.

ii. The inversion formula holds.

iii. The Parseval-Plancherel identities hold: $$\sum_{t\in\mathbb{R}/\mathbb{Z}}\left|f\left(t\right)\right|^{2}=\lim_{X\rightarrow\infty}\frac{1}{2X}\int_{-X}^{X}\left|\hat{f}\left(x\right)\right|^{2}dx$$

$$\sum_{t\in\mathbb{R}/\mathbb{Z}}f\left(t\right)\overline{g\left(t\right)}=\lim_{X\rightarrow\infty}\frac{1}{2X}\int_{-X}^{X}\hat{f}\left(x\right)\overline{\hat{g}\left(x\right)}dx$$

II.

i. What would be the analogue for $p\in\left[1,2\right]$ and $\ell^{p}\left(\mathbb{R}/\mathbb{Z}\right)$ of the fact that the Fourier transform is a continuous operator of norm $\leq1$ from $L^{p}\left(\mathbb{R}\right)$ to $L^{q}\left(\mathbb{R}\right)$, where $\frac{1}{p}+\frac{1}{q}=1$?

ii. How does the Hausdorff-Young inequality work in this situation, if at all?

III. Is there any reference I can turn to which covers the above issues of this variant of the Fourier transform, and which isn't completely co-opted by the terminology and agenda of Probability theory and characteristic functions?

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    $\begingroup$ In more conventional notation, you are considering the FT of the measure $\sum f(t)\delta_t$. I(iii) holds (by a straightforward calculation); it is usually called Wiener's theorem. $\endgroup$
    – Christian Remling
    Commented Dec 16, 2020 at 22:20
  • $\begingroup$ Question I(i) is unclear: obviously, the convergence won't be absolute if $f\notin\ell^1$, so order of summation matters, but there is no longer a distinguished natural ordering. $\endgroup$
    – Christian Remling
    Commented Dec 16, 2020 at 22:23
  • $\begingroup$ Where is a specific citation for this particular "Wiener's Theorem"? (There are a lot of results named after him, and more than one of those is relevant to my current research)? As for I(i), I suppose the main issue here is to properly identify what, if any, Banach space we can use to make sense of $\hat{f}$ for square-summable. Also, on a technical point, isn't your "more conventional notation" different than what I've given: your object doesn't take finite values on its "support"; whereas mine does? $\endgroup$
    – MCS
    Commented Dec 16, 2020 at 22:33
  • $\begingroup$ Also, is there a simple characterization of the image of $\ell^{1}\left(\mathbb{R}/\mathbb{Z}\right)$ under this Fourier transform? $\endgroup$
    – MCS
    Commented Dec 16, 2020 at 22:35
  • $\begingroup$ A formal remark: this should be denoted $\ell^p([0,1))$ rather than $\ell^p(\mathbb R/\mathbb Z)$, because the definition of the "Fourier transform" depends on this identification. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Dec 17, 2020 at 9:05

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