Let $\varepsilon>0$. I was interested in understanding the justification of defining the following function $\phi$ via its Fourier transform, satisfying the following properties:
(1) $\widehat{\phi}\in C^\infty(\mathbb{R})$
(2) $\widehat{\phi}(\xi)=1$ for $\xi\in[-\pi+\varepsilon,\pi-\varepsilon]$
(3) supp$(\widehat{\phi})\subset[-\pi-\varepsilon,\pi+\varepsilon]$
(4) $\widehat{\phi}$ goes from 1 to 0 monotonically on the intervals $[-\pi-\varepsilon,-\pi+\varepsilon]$ and $[\pi-\varepsilon,\pi+\varepsilon]$.
(5) $\underset{n\in\mathbb{Z}}\sum\widehat{\phi}(\xi-2\pi n)=1$ for every $\xi\in\mathbb{R}$.
The Fourier Transform convention used was $$\widehat{f}(\xi)=\int_\mathbb{R}f(x)e^{-ix\xi}dx.$$ I am sure there is no problem with it, but I don't quite understand how the construction goes. I would appreciate any explanation. Thank you very much!
In this answer I will carry out in detail the suggestions in Ian's comments.
Instead of constructing $\phi$, construct its Fourier transform $\Phi$. Let $$ b(\xi)=\begin{cases}c\,e^{-\tfrac{1}{1-\xi^2}} &\text{if }|\xi|\le1,\\0&\text{if }|\xi|>1, \end{cases}$$ where $c>0$ is chosen so that $\int_{-1}^1b(\xi)\,d\xi=1$. Next, let $\psi$ be the convolution of the characteristic function of the interval $[-\pi,\pi]$ with $\dfrac1\epsilon\,b\Bigl(\dfrac{\xi}{\epsilon}\Bigr)$. Then $\psi$ satisfies properties (1)-(4) as a function of $\xi$. Finally, observe that $\sum_{n\in\mathbb{Z}}\psi(\xi-2\pi n)$ is well defined, since only a finite number of summands are $\ne0$, and $>0$. Then define $$ \Phi(\xi)=\frac{\psi(\xi)}{\sum_{n\in\mathbb{Z}}\psi(\xi-2\pi n)} $$ and $\phi(x)$ as the inverse Fourier transform of $\Phi$.