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Math Help - Uniform Convergence

  1. #1
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    Uniform Convergence

    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)
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  2. #2
    Super Member girdav's Avatar
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    Re: Uniform Convergence

    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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