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I want to know how to approximate a general multivariable linear piecewise function (in both cases where it's continious but not smooth and when it's not continious nor smooth) $\mathbb{R}^{n} \rightarrow \mathbb{R}$ to a smooth continuous multivariable function $\mathbb{R}^{n} \rightarrow \mathbb{R}$

https://www.hindawi.com/journals/jam/2015/376362/

I found this paper but I am not sure if the method presented works for $n>1$

16π Cent
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You can use a convolution with a smooth function. My experience is that it's good to consider $f*g_\varepsilon$, where $g_\varepsilon(x)=g(x/\varepsilon)/\varepsilon^n$ and $g$ is a function with the following properties

  • $g(x)=0$ outside the unit ball
  • $g(x)\geq 0$ (this is not crucial)
  • $g$ is smooth
  • $\int_{\Bbb R^n} g(x)dx=1$

This is a random post about it that I found on MSE.

Mateo
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  • I'm guessing here x is a representation of more than 1 variable since we are in $\Bbb{R}^{n}$ ? and $\int_{\Bbb R^n} g(x)dx=1$ is the integral over the entire $\Bbb{R}^{n}$ domain? – 16π Cent Jul 09 '22 at 22:39
  • The answer is 'yes' to both your questions. – Mateo Jul 09 '22 at 22:40