I was watching a lecture from Stanford on the Fourier transform on distributions since following the textbook for my course I was able to follow through the theory and proofs but not able to actually calculate the Fourier transform of a tempered distribution.
In the textbook we defined the Fourier transform $F[\phi]$ on a test (Schwartz) function $\phi \in \mathscr{T}(\mathbb{R}^{n})$ by
$$F[\phi](\xi) := \int \phi(x) e^{i \langle \xi, x \rangle} dx.$$
So here is where the video differs from my textbook, namely, that the definition is
$$F[\phi](\xi) := \int_{- \infty}^{\infty} \phi(x) e^{-2 \pi i \xi x} dx.$$
So why are there two definitions? Is the second one just the one dimensional version and the video is considering $\mathbb{R}$ instead of $\mathbb{R}^{n}$?
And the Fourier transform on a tempered distribution $f$ operating on a test function $\phi$ by
$$\langle F[f], \phi \rangle := \langle f, F[\phi] \rangle.$$
The lecture defines it the same way.
Now, then we actually calculate the Fourier transform of the $\delta$ function such that
\begin{equation} \begin{aligned} \langle F[\delta], \phi \rangle &= \langle \delta, F[\phi] \rangle \\ & = F[\phi](0) \\ & = \int e^{i \langle 0, x \rangle}\phi(x) dx \\ & = \int 1 \cdot \phi(x) dx \\ & = \langle 1, \phi \rangle. \end{aligned} \end{equation}
Thus, $F[\delta] = 1$.
The lecture arrives at the same solution but using the second definition of the Fourier transform of a Schwartz function.
So now, when we consider the example of the $\delta_{a}$ function which acts on $\phi$ by assigning its value at $a$, my calculations using the definition from the book differ from that in the lecture namely, I receive
\begin{equation} \begin{aligned} \langle F[\delta_{a}], \phi \rangle &= \langle \delta_{a}, F[\phi] \rangle \\ & = F[\phi](a) \\ & = \int e^{i \langle a, x \rangle}\phi(x) dx \\ & = \langle e^{i \langle a, x \rangle}, \phi \rangle. \end{aligned} \end{equation}
Thus, $F[\delta_{a}] = e^{i \langle a, x \rangle}$. And in the video (if one uses the second definition) he finds $F[\delta_{a}] = e^{-2 \pi i a x}$. Even if I substitute $n = 1$ in my calculation I do not arrive at $F[\delta_{a}] = e^{-2 \pi i a x}$, but instead at $F[\delta_{a}] = e^{i a x}$.
So am I making mistakes somewhere or what did I miss?