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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?

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  • $\begingroup$ can't you 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? $\endgroup$ – mathworker21 Dec 5 '18 at 16:41
  • $\begingroup$ That is completely correct! Thanks, you can formulate this as a proper answer, thank you. $\endgroup$ – Darkwizie Dec 5 '18 at 19:28
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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.

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