Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, 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.

Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, 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.