I found this question in a previous test paper:

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/P3ba5.png

I'm having a few confusions:

1) Low pass filters are supposed to block signals below a certain frequency. Here that certain frequency is \$w_c=a\$. I'm not sure how to approach this problem. What would be the frequency of \$v(t) = e^{-at}\$?  Wolfram Alpha says that the Fourier transform of functions of the form \$e^{-at}\$ do not have any closed form. Also, Fourier series of the signal wouldn't exist since it's not periodic. So, how should I find the energy dissipated in the resistor? 

2) In the second part of the question, what exactly is meant by \$h(t)\$ and \$y(t)\$, any idea? 

Please note that I've read the answers to [How Fourier transform be able to deal with transients?](https://electronics.stackexchange.com/questions/317277/how-fourier-transform-be-able-to-deal-with-transients), but it doesn't really answer my question.