• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 composition lemma tommy1729 Ultimate Fellow     Posts: 1,620 Threads: 364 Joined: Feb 2009 04/21/2012, 05:04 PM Hi while trying to make a numerical method i stumbled upon a perhaps classical and/or old question : " composition lemma " f: R-> R g: R->R Let f [ g^(-1)[x] ] + g^(-1) [ f[x] ] = 2 g[x] (?) ==> (?) g^(2) [x] = f [x] regards tommy1729 tommy1729 Ultimate Fellow     Posts: 1,620 Threads: 364 Joined: Feb 2009 04/29/2012, 08:32 PM Let f [ g^(-1)[x] ] + g^(-1) [ f[x] ] = 2 g[x] replace x with g(x) hence we get for x e image g(x) f(x) + g^(-1)[ f[g(x)] ] = 2 g[g(x)] if f and g are real analytic then for all real x f(x) + g^(-1)[ f[g(x)] ] = 2 g[g(x)] Let g(g(x)) = f(x) + r(x)/2 hence f(x) + g^(-1)[ f[g(x)] ] = 2 f(x) + r(x) g^(-1)[ f(g(x)) ] = f(x) + r(x) thus r(x) = 0 for all x iff f(g(x)) = g(f(x)) which is the condition we needed. QED tommy1729 « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post Consistency in the composition of iterations Daniel 9 323 06/08/2022, 05:02 AM Last Post: JmsNxn Improved infinite composition method tommy1729 5 2,229 07/10/2021, 04:07 AM Last Post: JmsNxn Composition, bullet notation and the general role of categories MphLee 8 4,147 05/19/2021, 12:25 AM Last Post: MphLee Inverse super-composition Xorter 11 26,497 05/26/2018, 12:00 AM Last Post: Xorter [Collatz] Tommy's collatz lemma tommy1729 0 3,391 09/11/2014, 08:48 AM Last Post: tommy1729  composition of 3 functions. tommy1729 0 3,476 08/25/2014, 12:08 AM Last Post: tommy1729 lemma 1 tommy1729 2 7,773 06/30/2011, 11:47 AM Last Post: tommy1729 attracting fixed point lemma sheldonison 4 15,982 06/03/2011, 05:22 PM Last Post: bo198214 Uniqueness of tetration. Small lemma. Kouznetsov 9 15,781 04/14/2009, 01:20 AM Last Post: Kouznetsov

Users browsing this thread: 1 Guest(s) 