# Smoothing a function

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: $$f(x) = |x| \qquad f_n(x) = (x^{2}+n)^{1/2}$$

• 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. – reuns Feb 6 at 9:19
• 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). – Alexey Feb 6 at 15:15
• 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? – Malachi Wadas Feb 6 at 15:19