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Thread: Uniform Convergence

  1. #1
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    Uniform Convergence

    Let $\displaystyle g(x):\mathbb{R} \to \mathbb{R}$ be a function such that $\displaystyle \int_{-\infty}^\infty g(x) dx =1$. Show that for any continuous function $\displaystyle f:\mathbb{R} \to\mathbb{R}$,we ahve the sequence $\displaystyle g_n(x)=n\int_{-\infty}^\infty g(n(x-y))f(y)dx $converges uniformly on the compact sets of $\displaystyle \mathbb{R}$ to f(x)
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  2. #2
    Super Member girdav's Avatar
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    Re: Uniform Convergence

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