In my functional analysis class I saw the following problem: define Fourier transform (of a function) as $$\hat f(\xi)=\int\limits_{-\infty}^{\infty} f(x)e^{ix\xi}dx$$ then find the fourier transform of distribution (generalized function) $f(x)=\cos^2(8x)$ in $S'$ From all I know I started this way: $$\langle \hat f, \varphi\rangle=\langle f, \hat \varphi\rangle = \int\limits_{-\infty}^{+\infty} f(\xi)\int\limits_{-\infty}^{+\infty} \varphi(x)e^{ix\xi}dxd\xi=\int\limits_{-\infty}^{+\infty} \cos^2(8\xi)\int\limits_{-\infty}^{+\infty} \varphi(x)e^{ix\xi}dxd\xi$$
So now I/m thinking of what should I do next: should I just calculate this integral? I tried, no success. And when I was looking for some examples of Fourier transform in $S'$ it always was about some "known stuff" - like $\delta$-function, Heaviside function, $\mathcal P\frac 1 x$, etc. So may be I should use some fact like that?
Any hints are appreciated!
Let $f(x)=\cos^2(8x)$. Then, we have for $\phi\in \mathbb{S}$
$$\begin{align} \langle \mathscr{F}\{f\},\phi\rangle&=\langle f,\mathscr{F}\{\phi\}\rangle\\\\ &=\int_{-\infty}^\infty \cos^2(8x)\int_{-\infty}^\infty \phi(k)e^{ikx}\,dk\,dx\\\\ &=\pi \phi(0)+\frac1{2}\int_{-\infty}^\infty \cos(16x)\int_{-\infty}\phi(k) e^{ikx}\,dk\,dx\\\\ &=\pi \phi(0)+\frac\pi{2}\left(\phi(16)+\phi(-16)\right) \end{align}$$
Hence, we have in distribution
$$\mathscr{F}\{f\}(k)= \pi\delta(k)+\frac{\pi}2(\delta(k-16)+\delta(k+16))$$