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Math Help - uniform convergence series

  1. #1
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    uniform convergence series

    Suppose \sum f_n(x) converges uniformly on A and g :A \rightarrow B is a continuous function.
    Do we have any conclusion about the series \sum g(f_n(x)) ?
    ( A and B are subsets of real number)
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  2. #2
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    Quote Originally Posted by problem View Post
    Suppose \sum f_n(x) converges uniformly on A and g :A \rightarrow B is a continuous function.
    Do we have any conclusion about the series \sum g(f_n(x)) ?
    ( A and B are subsets of real number)
    No.

    \sum g(f_n(x)) might diverge or converge. Just take f_n(x)=2^{-n} and g(x)=1, in this case we have divergence. However, if g(x)=x we have convergence. So we need to know more about g
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    putnam120,if I change the g to be a uniformly continuous function,can I claim that \sum g(f_n) converges uniformly?

    Or it will be the case that convergence/uniform convergence of a series is not invariant under any continuous/uniform continuous function?
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  4. #4
    Senior Member Shanks's Avatar
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    Notice that g(x)=1 and g(x)=x are both uniformly continious function, the claim still can't be true.
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    So is there any properties that a function should have so that the uniform convergence of a series is invariant under the particular function?
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  6. #6
    Senior Member Shanks's Avatar
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    I think, the answer is negative.
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  7. #7
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    Quote Originally Posted by problem View Post
    putnam120,if I change the g to be a uniformly continuous function,can I claim that \sum g(f_n) converges uniformly?

    Or it will be the case that convergence/uniform convergence of a series is not invariant under any continuous/uniform continuous function?
    Going by what I think you mean you're misunderstanding your question:

    If f_n : A\rightarrow B and g: B \rightarrow \mathbb{R} are such that f_n \rightarrow f unif. on A and g is unif. cont. on B then g(f_n)\rightarrow g(f) unif. on A. Applying this result to the partial sums of a series, say \sum_{k=1}^{\infty } h_k = h (which converges unif.) we get that g(\sum_{k=1}^{n} h_k) \rightarrow g(h) uniformly. Thus unif. continuity does preserve unif. convergence. What you're asking however is convergence of the series given by \sum_{k=1}^{\infty } g(h_k). Do you see the difference?

    PS. I think this works for your original question:

    Let h_k,h : A \rightarrow B such that \sum_{k=1}^{\infty } h_k(x) = h(x) be a unif. conv. series of functions such that it converges absolutely for all x \in A, and g: B\rightarrow \mathbb{R} be unif. continous and \vert g(x) \vert \leq M \vert x \vert for all x\in B then \sum_{k=1}^{\infty } g(h_k) converges unif. on A and absolutely for all x\in A
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