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Fourier Series - Uniform and Pointwise Convergence Problem

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Please see the attached file for the f(x) function.

a) On the interval [a,b], does the sequence of functions converge pointwise? If yes, what is the limit function? Is the convergence uniform?

b) Answer the same three questions, but now let the function be defined on the real line.

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Solution Preview

fn(x)=0 if x<=n and fn(x)=x^2-n^2 if x>=n
(a) On the interval [a,b], fn(x) converges pointwise, then limit function is g(x)=0. The convergence is uniform.
Proof: For any e>0, we can find enough ...

Solution Summary

Uniform an pointwise convergence of a Fourier Series is investigated on an interval and across the entire real line.