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Math Help - uniformly convergent of sequence of integrals

  1. #1
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    uniformly convergent of sequence of integrals

    Let  f \in  C(\mathbb{R}) and M>0. Prove that the sequence of functions:
     f_n(z)=\frac{n}{2} \int^{z+\frac{1}{n}}_{z-\frac{1}{n}} f(y) dy is uniformly convergent to f in  [-M,M] .
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  2. #2
    Super Member girdav's Avatar
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    Re: uniformly convergent of sequence of integrals

    You can show, using the substitution u=nt in \frac n2\int_{-\frac 1n}^{\frac 1n}f(z+t)dt to get f_n(z)=\frac 12\int_{-1}^1f\left(z+\frac un\right)du. Now, use the fact that a continuous function on a compact set is uniformly continuous.
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