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Given a function $f(x)$ does there exist a sequence of smooth functions that $f_{n}(x) \to f(x)$ as $n \to 0$?

I am currently trying to smooth out a kink for the general function $x^{1/p}$.

An example of what I mean: $$f(x) = |x| \qquad f_n(x) = (x^{2}+n)^{1/2}$$

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  • $\begingroup$ You may want to use the convolution : let $\phi_n(x) = n e^{-\pi n^2 x^2}$ and $f_n(x) = f \ast \phi_n(x) = \int_{-\infty}^\infty f(x-y) \phi_n(y)dy$, additionally $f_n $ is analytic. $\endgroup$ – reuns Feb 6 at 9:19
  • $\begingroup$ Actually, it does not matter much which smooth functions you convolve with, but they should better be positive, with compact support, and their integral should be $1$ (or else a coefficient should be applied). $\endgroup$ – Alexey Feb 6 at 15:15
  • $\begingroup$ I see. Two questions: is there a way to find phi and what should I do if it doesn’t have a closed form expression? $\endgroup$ – Malachi Wadas Feb 6 at 15:19

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