Suppose converges uniformly on and is a continuous function.

Do we have any conclusion about the series ?

( and are subsets of real number)

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- January 7th 2010, 07:10 PMproblemuniform convergence series
Suppose converges uniformly on and is a continuous function.

Do we have any conclusion about the series ?

( and are subsets of real number) - January 7th 2010, 08:49 PMputnam120
- January 8th 2010, 08:19 AMproblem
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?(Wondering) - January 8th 2010, 08:49 AMShanks
Notice that g(x)=1 and g(x)=x are both uniformly continious function, the claim still can't be true.

- January 8th 2010, 08:55 AMproblem
So is there any properties that a function should have so that the uniform convergence of a series is invariant under the particular function?

- January 8th 2010, 08:59 AMShanks
I think, the answer is negative.

- January 8th 2010, 03:42 PMJose27
Going by what I think you mean you're misunderstanding your question:

If and are such that unif. on and is unif. cont. on then unif. on . Applying this result to the partial sums of a series, say (which converges unif.) we get that uniformly. Thus unif. continuity does preserve unif. convergence. What you're asking however is convergence of the series given by . Do you see the difference?

PS. I think this works for your original question:

Let such that be a unif. conv. series of functions such that it converges absolutely for all , and be unif. continous and for all then converges unif. on and absolutely for all