Uniform Convergence

**problem** · July 10th 2012, 10:18 PM

Let $g(x):\mathbb{R} \to \mathbb{R}$ be a function such that $\int_{-\infty}^\infty g(x) dx =1$ . Show that for any continuous function $f:\mathbb{R} \to\mathbb{R}$ ,we ahve the sequence $g_n(x)=n\int_{-\infty}^\infty g(n(x-y))f(y)dx$ converges uniformly on the compact sets of $\mathbb{R}$ to f(x)

**girdav** · July 11th 2012, 12:57 AM

First, use the substitution $t=n(x-y)$ in the integral (the integration is with respect to $y$ ). I think we have to assume $f$ bounded, because if $tg(t)$ is not integrable (for example when $g(t)=\frac 1{\pi(1+t^2)}$ , and $f(t)=t$ , we have some problems.

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