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Real analysis

We have just finished up integration and are done with a first course in analysis, so chapters 1-6 of Rudin. We are also using the Ross and Morrey/Protter book. Please answer question fully and clearly explaining every step. Any solution short of perfect is useless to me. So if you are not 100% sure whether your answer is right, then please do not answer. The same problem is also attached as a word document with all the symbols. ****************************************************** Let f: [a,b] --> R an integrable function. Prove that:
i) lim *S* f(x) cos(nx) dx = 0
ii) lim *S* f(x) sin(nx) dx = 0.

where lim is n as it approaches plus infinity (it is not specified so I believe the default when only n is listed is to plus infinity, I may be mistaken), *S* is my notation for the integral taken from a to b.


Solution Preview

This is not necessarily true if f is merely integrable. It must be absolutely integrable on [a,b]. For example, if a=0 and b=infinity
then *S* cos(x^2) cos(sx) = (1/2)SQRT(PI/2)[cos(s^2/4)+sin(s^2/4] which doesn't go to zero as s->infinity.
If a and b are finite then f integrable implies f absolutely integrable on [a,b]... so we're cool there.
You should have a Lemma something ...

Solution Summary

This is a proof regarding limits of integrable functions.