Bruno Admin
Posts : 184 Join date : 20090915 Age : 30 Location : the infinite, frictionless plane of uniform density
 Subject: [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 

Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 34 Location : Right behind you
 Subject: 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 {mMnN 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. 

Bruno Admin
Posts : 184 Join date : 20090915 Age : 30 Location : the infinite, frictionless plane of uniform density
 Subject: 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! Where do you use the continuity of m and n? 

Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 34 Location : Right behind you
 Subject: Re: [SOLVED] Sum of periodic functions Sun Nov 08, 2009 10:21 pm  
 suppose that m_kMn_kN is convergent to x_0 so you get m(x_0)+n(x_0)= lim {m(m_kMn_kN)+n(m_kMn_kN)}=lim{m(n_kN)+n(m_kM)}=m(\infty)+n(\infty)=const. 

Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 34 Location : Right behind you
 
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 Subject: Re: [SOLVED] Sum of periodic functions  
 
