Analytic functions fn s.t. ((fn) o...o (fn))(x) = exp(x)
David L Dowe
I understand and vaguely recall that in about 1955 a mathematician in
Adelaide, Australia gave an example of a C2 (twice continuously-differentiable)
function f2 such that ((f2 o f2))(x) = exp(x) .
It seems intuitively clear that for any natural number n there exists
a real-valued (monotone) C_infinity function fn s.t. (fn o...o fn)(x) = exp(x)
where this denotes composition n times .
The C2 construction is clever, but (in hindsight) not that difficult.
If we want to generalise exp (and its inverse, log) to fractional/infinitesimal
operators, then surely we want to do this in a way which is C_infinity.
Question 1: Are such analytic functions fn unique?
(I suspect, unfortunately, not.)
The next question is not independent of the answer to Question 1.
If we define G to be such that for all natural numbers n and for all real x
G(1/n, x) = fn(x)
G( 0, x) = x
and for all natural numbers m
G(m/n, x) = ((fn o...o fn))(x) {where this denotes composition m times}
then
Question 2: Can we (uniquely) define G: R+U{0} xR -> R such that G(r, x) is the
pointwise convergence of G(q1, x), G(q2, x), ... where q1, q2, ... -> r ?
Assuming monotonicity of G(r, x) for r>0, we could (being a little careful
about domains for x) then define G(r, x) for r < 0
(as log, the inverse of exp, is also of interest) .
(Battling on) Question 3:
Such a G should be C_infinity in x (although we anticipate a problem for x<=0
when r < 0). One presumes likewise in r.
Can we write an explicit power series representation of G?
Thank you (in anticipation) for any constructive comments, and regards.
(Dr.) David Dowe, Dept of Comp Sci, Monash Univ, Clayton, Vic 3168, Australia.
(CS net:) d...@bruce.cs.monash.edu.au
Arthur Rubin
> I understand and vaguely recall that in about 1955 a mathematician in
>Adelaide, Australia gave an example of a C2 (twice continuously-differentiable)
>function f2 such that ((f2 o f2))(x) = exp(x) .
> It seems intuitively clear that for any natural number n there exists
>a real-valued (monotone) C_infinity function fn s.t. (fn o...o fn)(x) = exp(x)
> where this denotes composition n times .
I asked that question a few weeks ago, quoting Feynmann of Caltech around
1976. C_infinity is easy, real-analytic is not, and I haven't read the
references I was given yet to determine the precise answer.
>The C2 construction is clever, but (in hindsight) not that difficult.
>If we want to generalise exp (and its inverse, log) to fractional/infinitesimal
>operators, then surely we want to do this in a way which is C_infinity.
>Question 1: Are such analytic functions fn unique?
>(I suspect, unfortunately, not.)
C_infinity is NOT analytic.
>The next question is not independent of the answer to Question 1.
>If we define G to be such that for all natural numbers n and for all real x
>G(1/n, x) = fn(x)
>G( 0, x) = x
>and for all natural numbers m
>G(m/n, x) = ((fn o...o fn))(x) {where this denotes composition m times}
>then
>Question 2: Can we (uniquely) define G: R+U{0} xR -> R such that G(r, x) is the
>pointwise convergence of G(q1, x), G(q2, x), ... where q1, q2, ... -> r ?
>Assuming monotonicity of G(r, x) for r>0, we could (being a little careful
>about domains for x) then define G(r, x) for r < 0
>(as log, the inverse of exp, is also of interest) .
>(Battling on) Question 3:
>Such a G should be C_infinity in x (although we anticipate a problem for x<=0
>when r < 0). One presumes likewise in r.
>Can we write an explicit power series representation of G?
I assume you want G(x+y,z) to be G(x,G(y,z)) (where defined), even if it
doesn't follow from the above conditions. Clearly G must be monotone
increasing in each variable; hence consider the function g such that
x = G(g(x),0). With some research, we can find that
g(0) = 0
g is monotone increasing
g maps R to (-1,infinity)
and
g(exp(x)) = g(x) + 1.
(-1)
Conversely, if such a g exists, then G(x,y) = g (x + g(y)) satisfies all
the conditions you have for G.
References:
wven...@spam.maths.adelaide.edu.au (Bill Venables) cites:
Kneser, H., Reele analytische L\"osungen der Gleichung
$\phi(\phi(x))=e^x$. J. f. reine angew. Math. v. 187, (1950) 56--67.
Szekeres, G., Fractional iteration of exponentially growing functions. J.
Australian Math. Soc. v. 2, (1962) 301-320.
--
Arthur L. Rubin
216-...@mcimail.com 7070...@compuserve.com art...@pnet01.cts.com (personal)
a_r...@dsg4.dse.beckman.com (work) Beckman Instruments/Brea
My opinions are my own, and do not represent those of my employer.
Matthew P Wiener
>>function f2 such that ((f2 o f2))(x) = exp(x) .
>> It seems intuitively clear that for any natural number n there exists
>>a real-valued (monotone) C_infinity function fn s.t. (fn o...o fn)(x) = exp(x)
>> where this denotes composition n times .
>
>I asked that question a few weeks ago, quoting Feynmann of Caltech around
>1976. C_infinity is easy, real-analytic is not, and I haven't read the
>references I was given yet to determine the precise answer.
Analytic is possible.
>>[G(x,y) is to be a one-parameter family of compositional factors for exp,
>> ie, G(0,y)=exp(y), y |--> G(1/n,y) iterated n times is exp(y), etc.]
>I assume you want G(x+y,z) to be G(x,G(y,z)) (where defined), even if it
>doesn't follow from the above conditions. Clearly G must be monotone
>increasing in each variable; hence consider the function g such that
>x = G(g(x),0). With some research, we can find that
>g(0) = 0
>g is monotone increasing
>g maps R to (-1,infinity)
>and
>g(exp(x)) = g(x) + 1.
> (-1)
>Conversely, if such a g exists, then G(x,y) = g (x + g(y)) satisfies all
>the conditions you have for G.
Call the inverse function h. Then you want solutions to h(z+1)=exp(h(z)).
Entire functions satisfying such were constructed in C Horowitz "Iterated
Logarithms of Entire Functions" ISRAELI JOURNAL OF MATHEMATICS, v29, n1,
pp31-42 (1978).
Horowitz proved that given a sequence of complex numbers <a_i:i=0,...> st
a_i=exp(a_j) (where j=i+1--maybe this should be an ASCII convention?) then
there exists a unique sequence of entire functions <f_i:i=0,...> st f_i=
exp(f_j), f_i(0)=a_0, f'_0(0)=1, and lim f''_n/f'_n=0 uniformly on compact
subsets of C.
Now consider fixed points of exp. There are infinitely many, by Picard.
Let u be one such. Consider the above result with a_i=u, all i. Since
<f_j(uz):i=0,...> satisfies the above, by uniqueness, f_j(uz)=f_i(z)=
exp(f_j(z)). So we have solutions to f(uz)=exp(f(z)). Given such an f,
h(w)=f(exp(uw)) is entire and satisifies h(w+1)=exp(h(w)).
With g the inverse of h, we can write Q(z)=h(g(z)+.5) which is entire and
see that Q(Q(z))=exp(z), as mentioned in > above. If we write F for the
inverse of f, we have Q(z)=f(sqrt(u)*F(z)).
I believe that all the Q's generated by this procedure are the same. The
different choices of exp fixed point are taken at different values of f,
again by Picard, and so the uniqueness results of Horowitz should apply
to translate from one f into another. So is Q unique for its functional
equation among entire functions, given initial conditions and some other
mild restrictions? Ditto for the other fractional iterates and for h.
>References: wven...@spam.maths.adelaide.edu.au (Bill Venables) cites:
>Kneser, H., Reele analytische L\"osungen der Gleichung
>$\phi(\phi(x))=e^x$. J. f. reine angew. Math. v. 187, (1950) 56--67.
>Szekeres, G., Fractional iteration of exponentially growing functions. J.
>Australian Math. Soc. v. 2, (1962) 301-320.
--
-Matthew P Wiener (wee...@libra.wistar.upenn.edu)