[FOM] 605: Integer and Real Functions (Harvey Friedman)

[Joe Shipman]

I should point out that the problem here, of extending the "tower" function to the reals in a satisfactory way, is not simply one of "fast growth". It would be enough to find a function such that f(f(x))= sin(x), for example.

Consider the sequence of functions
f_1(x)=sin(x)
f_i+1(x)=sin(f_i(x))

These all look like sine waves with decreasing amplitude. For each x between 0 and pi/2, the sequence of values seems to converge "nicely" to zero, but apparently not nicely enough that there is any "natural" way of making it continuous.

Or is there?

-- JS


Sent from my iPhone

> On Aug 28, 2015, at 12:27 AM, Joe Shipman <joeshipman at aol.com> wrote:
> 
> What do you mean "none that are holomorphic"?
> 
> Is there actually a theorem of the following form?
> 
> "there is no real function f(x) such that 
> (1) there is a nonempty interval (a,b) in the reals such that for all x in (a,b), f(f(x))=e^x
> (2) for some open set W in the complex plane that includes  the reals in (a,b), there is a holomorphic extension of f defined on W."
> 
> -- JS
> 
> 
> 
> 
> Sent from my iPhone
> 
>> On Aug 27, 2015, at 1:50 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
>> 
>> Harvey Friedman wrote:
>>> So what we would like to have is a TWO VARIABLE functional equation with a initial condition that uniquely determines factorial on R|>=0.
>> 
>> One thing to try, since you have a two-variable functional equation for exponentiation, is to use the fact that the gamma function is the Mellin transform of exp(-x).  Of course then one has to come up with a "fundamental purpose" for the Mellin transform.  Perhaps the fact about the Mellin transform that is closest in spirit to what you are trying to do here is that if X and Y are two independent random variables, then the Mellin transform of their products is equal to the product of the Mellin transforms of X and Y.
>> 
>>> QUESTION: Find some natural classes of functions, X, such that for all
>>> f in X there is a (unique) g in X such that f = gog.
>>> 
>>> Horrible things seem to happen when you look at, e.g.,
>>> 
>>> 2^n = g(g(n)) on N
>>> e^x = h(h(x)) on R|>=n.
>>> 
>>> The intuitive idea is clear. Let f(x) be the result of doing something
>>> to x. Then g(x) should be doing it half way. When, there, and how does
>>> this make sense?
>> 
>> Well, there's a lot of literature on this subject, e.g.,
>> 
>> http://reglos.de/lars/ffx.html
>> 
>> I'm not sure that there's anything that you'd find very satisfying, though.  For example, there do exist moderately well-behaved solutions to e^x = h(h(x)) but none that are holomorphic, so it's not clear that any of the solutions are "fundamental."
>> 
>> Tim
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