Efficient way to do a Fourrier Transform like operation

Question

Suppose we have two functions $f,g:[0,\infty) \rightarrow [0,\infty)$. Then one can use Fast Fourrier Transforms to quickly compute $\int_0^t f(t-s) g(s) \, ds$ for $t$ in some range of values $[0,T]$ this can be done for example in Matlab using ifft(fft(f).*fft(g)).

Now let $M \in [0,\infty)$ be some number and assume we want to compute the integral $\int_0^M f(t-s) g(s) \, ds$ for $t$ in some range $[M,T]$. Is there something similar we can do?

Answer 1

You can just define g to be $0$ on the range $[M,T]$ so that the integral is equal to the integral over the range $[0,T]$ and then you can apply the method you mentioned first.