1) Does a convolution of an integrable function and an unintegrable function exist?
2) If the answer for the first question is possible, then: suppose a function $f(x)$ is integrable, and denote by $g(x)=\sum_{|n|\geq N} e^{inx}$ for a given $N$. Can we use the "convolution theorem" in this case? (this is a sort of "high pass filter")
It seems to me that in order to do this, we need to change the order of summation and integration, but $g$ diverges.