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I have a basic question which I am confused about. If I have two functions $f(x)$, and $g(x)$, where $g(x)$ has a compact support say $[-M,M]$. Can I always say that I can just consider the integral over $[-M,M]$. Namely,

$$ f*g(x)= \int_{-M}^{M}f(x-y)g(y)dy= \int_{-M}^{M}f(y)g(x-y)dy $$ I am worried that what if $f$ blows up outside the support of $g(x)$. I know that if $f(x)$ is bounded, then there is no problem what that statement. Thanks!

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The first equality is correct but the second one is not. The last integral should be changed to $\int_{x-M}^{x+M} f(y)g(x-y)\, dy$.

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  • $\begingroup$ Oh I see. Thanks! I am wondering whether the first equality always holds. What if $f(x)$ blows up near the compact support of $g(x)$ $\endgroup$ – Demha Feb 4 '19 at 18:00
  • $\begingroup$ Just to reword what I mean. In order to use $f*g(x)= \int_{-M}^{M}f(x-y)g(y)dy= \int_{-M}^{M}f(y)g(x-y)dy$, I need to make sure that $f(x)$ is bounded right? Thanks again! $\endgroup$ – Demha Feb 4 '19 at 18:20

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