Let $$f(t)=\begin{cases}0,&|t|\geqslant 1\\1-|t|,&|t|<1\end{cases}.$$ Compute the Fourier transform of $(f(t))^{2}$.
So I am a beginner with Fourier transformations and got stuck with this question. My attempt was to try and use convolution in some way as $f(t) = \widehat{g(x)}$ but I am unable to figure out how to do this. This is the final part of a course in complex analysis so the information I have available is not particularly detailed. Any help would be appreciated.
$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\on}[1]{\operatorname{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} \on{f}\pars{t} & \equiv \bracks{-1 < t < 0}\pars{1 + t} \\[2mm] & +\,\, \bracks{0 < t < 1}\pars{1 - t} \\[5mm] & = \bracks{\verts{t} < 1}\pars{1 - \verts{t}} \\[5mm] \implies \on{f}^{2}\pars{t} & = \bracks{\verts{t} < 1} \pars{1 - \verts{t}}^{\,2} \end{align}
\begin{align} \varphi\pars{\omega} & \equiv \int_{-\infty}^{\infty}\on{f}^{2}\pars{t} \expo{-\ic\omega t}\,\dd t = \int_{-1}^{1}\pars{1 - \verts{t}}^{\,2} \expo{-\ic\omega t}\,\dd t \\[5mm] & = 2\int_{0}^{1}\pars{1 - t}^{\,2} \cos\pars{\omega t}\,\dd t \\[5mm] & = \bbx{4\,{\omega - \sin\pars{\omega} \over \omega^{3}}} \\ &\ \end{align}