Following M. Ruzhansky and V. Turunen's book [Pseudo-Differential Operators and Symmetries](http://www.math.nagoya-u.ac.jp/%7Erichard/teaching/s2017/Ruzhansky_Turunen.pdf), in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (paraphrasing):



> 
> **Definition 3.1.25 (Periodic Distributions)** The space of periodic distributions is defined as dual space $\mathcal{D}'(\mathbb{T}^n) = \mathcal{L}(C^{\infty}(\mathbb{T}^n),\mathbb{C})$ (i.e. of continuous linear operators $u:C^{\infty}(\mathbb{T}^n) \to \mathbb{C}$), where we write $u(\varphi) = \langle u , \varphi\rangle$ for any $\varphi\in C^{\infty}(\mathbb{T}^n)$. Furthermore, for any $\psi\in C^\infty(\mathbb{T}^n)$ or $L^p(\mathbb{T}^n)$, define the associated distribution to $\psi$ in the canonical sense, i.e. $u\_{\psi} \in \mathcal{D}'(\mathbb{T}^n)$ where for any $\varphi \in C^\infty(\mathbb{T^n})$
> \begin{align}u\_{\psi}(\varphi) = \langle u\_{\psi}, \varphi\rangle=\int\_{\mathbb{T}^n}\psi(x) \varphi(x) dx.\end{align}
> 
> 
> 


Given the above definition, as with standard distributions, we can define a Fourier transform of periodic distributions with respect to Fourier transforms of functions.



> 
> **Definition 3.1.8 (Periodic Fourier Transform)** Let $\mathcal{F}\_{\mathbb{T}^n} : C^{\infty}(\mathbb{T^n}) \to \mathcal{S}(\mathbb{Z}^n) $ be defined as
> $$ \begin{align} (\mathcal{F}\_{\mathbb{T}^n}f)(\xi):=\int\_{\mathbb{T}^n} f(x) e^{-2\pi i x \cdot \xi} dx \end{align}$$
> for $f \in C^{\infty}({\mathbb{T}^n})$, and its inverse be defined as
> $$\begin{align} (\mathcal{F}\_{\mathbb{T}^n}^{-1}h)(x) = \sum\_{\xi \in \mathbb{Z}^n} h(\xi) e^{2\pi i x \cdot \xi}\end{align}$$
> for $h\in\mathcal{S}({\mathbb{Z}^n})$.
> 
> 
> 



> 
> **Definition 3.1.27 (Fourier Transform of Periodic Distributions)** For any $u\in\mathcal{D}'(\mathbb{T}^n)$, define the Fourier transform $\mathcal{F}\_{\mathbb{T}^n}:\mathcal{D}'(\mathbb{T}^n) \to \mathcal{S}'(\mathbb{Z}^n)$ as
> $$\begin{align}\langle \mathcal{F}\_{\mathbb{T}^n} u, \varphi \rangle = \langle u, \mathcal{F}\_{\mathbb{T}^n}^{-1} \varphi(-\cdot) \rangle\end{align} \tag{1}$$
> for any $\varphi \in C^{\infty}(\mathbb{T}^n)$
> 
> 
> 


**QUESTION**
------------


My question is, is the definition of the Fourier transform on periodic distributions, as given in Definition 3.1.27, **correct**? I feel that the definition should not involve the inverse Fourier transform, but instead be defined as
$$\begin{align} \langle \mathcal{F}\_{\mathbb{T}^n} u, \varphi \rangle = \langle u, \mathcal{F}\_{\mathbb{T}^n} \varphi \rangle. \tag{2}
\end{align}$$


**Reason**
----------


The reason I feel that this should be the case is because, if we consider the periodic Dirac distribution at zero, defined as $\delta\_0 \in \mathcal{D}'(\mathbb{T}^n)$ where
$$
\begin{align}\langle \delta\_0, \varphi\rangle = \varphi(0)\end{align}
$$
for any $\varphi \in C^{\infty}(\mathbb{T}^n)$, then its Fourier transform using $(1)$ gives
$$
\begin{align}
\underbrace{\langle \mathcal{F}\_{\mathbb{T}^n} \delta\_0, \varphi \rangle}\_{\in \mathbb{C}} &= \langle \delta\_0, \mathcal{F}\_{\mathbb{T}^n}^{-1} \varphi (-\cdot) \rangle \\
&= \mathcal{F}\_{\mathbb{T}^n}^{-1} \varphi (0) \\
&= \sum\_{\xi \in \mathbb{Z}^n} \varphi(\xi),
\end{align}
$$
where the right hand side is infinite since $\varphi$ is periodic. So this doesn't make sense. If we instead use $(2)$, then we obtain the following:
$$
\begin{align}
\underbrace{\langle \mathcal{F}\_{\mathbb{T}^n} \delta\_0, \varphi \rangle}\_{\in \mathbb{C}} &= \langle \delta\_0, \mathcal{F}\_{\mathbb{T}^n} \varphi \rangle \\
&= (\mathcal{F}\_{\mathbb{T}^n} \varphi) (0)\\
&= \int\_{\mathbb{T}^n} \varphi(x) dx \\
&= \langle 1, \varphi \rangle
\end{align}
$$
i.e. $\mathcal{F}\_{\mathbb{T}^n} \delta\_0 = 1$ in the sense of periodic distributions, as we would expect.



Fourier transform of periodic distributions
Actually, the definition you gave in the post differs from the one in the book. The test function $\varphi$ should lie in $\mathcal S(\mathbb Z^n)$, not in $C^\infty(\mathbb T^n)$, since the $\mathcal F\_{\mathbb T^n}$ maps the second of these spaces into the first, so for $\varphi\in C^\infty(\mathbb T^n)$ the expression $\mathcal F\_{\mathbb T^n}^{-1}\varphi$ is undefined, as you correctly noticed. So, the correct formula would be
$$
\langle\mathcal F\_{\mathbb T^n}u,\varphi \rangle=\langle u,\iota\,\circ\,\mathcal F\_{\mathbb T^n}^{-1}\varphi \rangle,
$$
where $\iota(f)(x)=f(-x)$ and $\varphi \in \mathcal S(\mathbb Z^n)$.


