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I was given as homework a question related with this "smoothing" function: ${ F }_{ \delta }(x)=\frac { 1 }{ 2\delta } \int _{ -\delta }^{ \delta }{ f(x+t)dt } $

Where $f(x)$ is a continuous function.

Does it have some formal name? I'm trying to read some more about it but I wasn't able to find useful info.

Thanks!

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  • $\begingroup$ It's the continous form of a moving average, see en.wikipedia.org/wiki/Moving_average. $\endgroup$
    – fgp
    Commented May 11, 2013 at 16:32
  • $\begingroup$ In the frequency domain, the operator which takes $f$ to $F_\delta$ is called a "brick wall filter", since it simply removes all frequences greater than $2\pi\delta$ from $f$. $\endgroup$
    – fgp
    Commented May 11, 2013 at 16:34

1 Answer 1

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If you generalize it a bit, you get a well known and much used construction. The integral can be written the form $$F_\delta(x)=\frac1\delta\int_{-\infty}^{\infty}f(x-t)\varphi\Bigl(\frac t\delta\Bigr)\,dt$$ where, in this case, $$\varphi=\frac12\chi_{[-1,1]}$$ (where, in general, $\chi_A$ is the function which takes the value $1$ in the set $A$ and $0$ outside it). You can replace $\varphi$ by any nonnegative function which vanishes outside the interval $[-1,1]$ and has $\int_{-\infty}^\infty\phi(t)\,dt=1$. Then $F_\delta\to f$ when $\delta\to0$. This is most useful when $\varphi$ is infinetely differentiable, for then $F_\delta$ too becomes infinitely differentiable.

The function $\varphi$ is called a mollifier, and the process of moving from $f$ to $F_\delta$ is called mollification. It is a much used technique in real analysis.

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