# Efficient way to do a Fourrier Transform like operation

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?

• 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? – mathworker21 Dec 5 '18 at 16:41
• That is completely correct! Thanks, you can formulate this as a proper answer, thank you. – Darkwizie Dec 5 '18 at 19:28

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.