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 [SOLVED] Sum of periodic functions

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Bruno
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PostSubject: [SOLVED] Sum of periodic functions   Sun Nov 08, 2009 9:18 pm

Let m(x) and n(x) be two continuous functions R-->R. Suppose m(x) is periodic with period M and n(x) is periodic with period N. Show that m(x)+n(x) is periodic if and only if M/N is rational.


Last edited by Bruno on Wed Nov 11, 2009 5:51 pm; edited 1 time in total
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Mohammad
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PostSubject: Re: [SOLVED] Sum of periodic functions   Sun Nov 08, 2009 9:54 pm

Sol: simply solved
if M/N be an irrational number then the set {mM-nN where m,n\in \mathbb{N}} is dense in \mathbb{R}, so you then get functions m and n just differs by a constant and more M=N which is of course a contradiction. bounce
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Bruno
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PostSubject: Re: [SOLVED] Sum of periodic functions   Sun Nov 08, 2009 10:15 pm

It's good, I think you have the general idea, but you don't give much details! Question Where do you use the continuity of m and n?
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Mohammad
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PostSubject: Re: [SOLVED] Sum of periodic functions   Sun Nov 08, 2009 10:21 pm

suppose that m_kM-n_kN is convergent to x_0 so you get
m(x_0)+n(x_0)= lim {m(m_kM-n_kN)+n(m_kM-n_kN)}=lim{m(-n_kN)+n(m_kM)}=m(\infty)+n(\infty)=const.
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Mohammad
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PostSubject: Re: [SOLVED] Sum of periodic functions   Sun Nov 08, 2009 10:27 pm

Very Happy , by the way, your proof today was just awesome bounce
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