half operator problem -- wondering if this is a discussed topic
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From: Lee Davidson on 9 Jun 2010 22:39 I tried some Google searches, but don't know what keywords to use. Here's the problem: given a function f mapping the reals into the reals, find g mapping the reals into the reals such that f(x) = g(g(x)) for any real x. For f(x) = ax+b this is easy. For f(x) = x^b for any real b this is easy. For f(x) = x^2 + 1 this looks difficult. Also, for f(x) = exp(x) this looks difficult. Is this discussed anywhere? And where, and what are appropriate keywords for a web search?
From: achille on 9 Jun 2010 23:18 On Jun 10, 10:39Â am, Lee Davidson <l...(a)meta5.com> wrote: > I tried some Google searches, but don't know what keywords to use. > > Here's the problem: given a function f mapping the reals into the > reals, find g mapping the reals into the reals such that f(x) = > g(g(x)) for any real x. > > For f(x) = ax+b this is easy. > > For f(x) = x^b for any real b this is easy. > > For f(x) = x^2 + 1 this looks difficult. > > Also, for f(x) = exp(x) this looks difficult. > > Is this discussed anywhere? And where, and what are appropriate > keywords for a web search? "functional equation" is one possible keyword. In particular, Wiki's entry on "functional equation": http://en.wikipedia.org/wiki/Functional_equation refers to a paper by M. Kuczma "On the functional equation Ï^n(x) = g(x)". which give a partial solution to the problem of finding continuous function f(x) which satisfies f(f(f..f(x))) = g(x).
From: Robert Israel on 10 Jun 2010 03:24 On Wed, 9 Jun 2010 19:39:54 -0700 (PDT), Lee Davidson wrote: > I tried some Google searches, but don't know what keywords to use. > > Here's the problem: given a function f mapping the reals into the > reals, find g mapping the reals into the reals such that f(x) = > g(g(x)) for any real x. > > For f(x) = ax+b this is easy. > > For f(x) = x^b for any real b this is easy. > > For f(x) = x^2 + 1 this looks difficult. > > Also, for f(x) = exp(x) this looks difficult. > > Is this discussed anywhere? And where, and what are appropriate > keywords for a web search? See e.g. the sci.math threads "Problem: finding f s.t. f(f(x)) = g(x) for given g" from 1998 <http://groups.google.ca/group/sci.math/browse_thread/thread/a3a12669f6dfab8c> "f(f(x)) = g(x)" and "No f:R -> R with f(f(x)) = g(x)" from 1999 <http://groups.google.ca/group/sci.math/browse_thread/thread/bb117de2a0c09022> <http://groups.google.ca/group/sci.math/browse_thread/thread/bceb6c01e6e78ab6> and "-- g o g = f" from 2009 <http://groups.google.ca/group/sci.math/browse_thread/thread/6b8330859a83630d> -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Lee Davidson on 10 Jun 2010 23:05 On Jun 10, 12:24 am, Robert Israel <isr...(a)math.MyUniversitysInitials.ca> wrote: > On Wed, 9 Jun 2010 19:39:54 -0700 (PDT), Lee Davidson wrote: > > I tried some Google searches, but don't know what keywords to use. > > > Here's the problem: given a function f mapping the reals into the > > reals, find g mapping the reals into the reals such that f(x) = > > g(g(x)) for any real x. > > > For f(x) = ax+b this is easy. > > > For f(x) = x^b for any real b this is easy. > > > For f(x) = x^2 + 1 this looks difficult. > > > Also, for f(x) = exp(x) this looks difficult. > > > Is this discussed anywhere? And where, and what are appropriate > > keywords for a web search? > > See e.g. the sci.math threads > "Problem: finding f s.t. f(f(x)) = g(x) for given g" from 1998 > <http://groups.google.ca/group/sci.math/browse_thread/thread/a3a12669f...> > "f(f(x)) = g(x)" and "No f:R -> R with f(f(x)) = g(x)" from 1999 > <http://groups.google.ca/group/sci.math/browse_thread/thread/bb117de2a...> > <http://groups.google.ca/group/sci.math/browse_thread/thread/bceb6c01e...> > and "-- g o g = f" from 2009 > <http://groups.google.ca/group/sci.math/browse_thread/thread/6b8330859...> > > -- > Robert Israel isr...(a)math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada- Hide quoted text - > > - Show quoted text - Thanks all. Now, it has occurred to me that one approach would be through power series, iterated. Supposing g(x) = a_0 + a_1 x + a_2 x^2 ... Let f(x) = b_0 + b_1 x + b_2 x^2 ... Then f(x) = g(g(x)) = a_0 + a_1 g(x) + a_2 g(x)^2 ... Then substitute g's power series and f's power series and solve the simultaneous equations. Aaaaarrrrgh!
From: Pubkeybreaker on 11 Jun 2010 07:05
On Jun 10, 11:05 pm, Lee Davidson <l...(a)meta5.com> wrote: > On Jun 10, 12:24 am, Robert Israel > > > > > > <isr...(a)math.MyUniversitysInitials.ca> wrote: > > On Wed, 9 Jun 2010 19:39:54 -0700 (PDT), Lee Davidson wrote: > > > I tried some Google searches, but don't know what keywords to use. > > > > Here's the problem: given a function f mapping the reals into the > > > reals, find g mapping the reals into the reals such that f(x) = > > > g(g(x)) for any real x. > > > > For f(x) = ax+b this is easy. > > > > For f(x) = x^b for any real b this is easy. > > > > For f(x) = x^2 + 1 this looks difficult. > > > > Also, for f(x) = exp(x) this looks difficult. > > > > Is this discussed anywhere? And where, and what are appropriate > > > keywords for a web search? > > > See e.g. the sci.math threads > > "Problem: finding f s.t. f(f(x)) = g(x) for given g" from 1998 > > <http://groups.google.ca/group/sci.math/browse_thread/thread/a3a12669f....> > > "f(f(x)) = g(x)" and "No f:R -> R with f(f(x)) = g(x)" from 1999 > > <http://groups.google.ca/group/sci.math/browse_thread/thread/bb117de2a....> > > <http://groups.google.ca/group/sci.math/browse_thread/thread/bceb6c01e....> > > and "-- g o g = f" from 2009 > > <http://groups.google.ca/group/sci.math/browse_thread/thread/6b8330859....> > > > -- > > Robert Israel isr...(a)math.MyUniversitysInitials.ca > > Department of Mathematics http://www.math.ubc.ca/~israel > > University of British Columbia Vancouver, BC, Canada- Hide quoted text - > > > - Show quoted text - > > Thanks all. Now, it has occurred to me that one approach would be > through power series, iterated. Supposing > > g(x) = a_0 + a_1 x + a_2 x^2 ... > > Let f(x) = b_0 + b_1 x + b_2 x^2 ... > > Then f(x) = g(g(x)) = a_0 + a_1 g(x) + a_2 g(x)^2 ... > > Then substitute g's power series and f's power series and solve the > simultaneous equations. > > Aaaaarrrrgh!- Hide quoted text - > > - Show quoted text - See: B. Reznick http://mathforum.org/kb/thread.jspa?forumID=253&threadID=567620&messageID=1693646 |