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In this question I said with an analytic function $g(z) = z+\sum_{n=2}^\infty b_n z^n$ the equation $$f(f(z)) = g(z)$$ has a formal solution $$f(z) = \sum_{n=1}^\infty c_n z^n, \qquad c_1 = 1, \qquad c_m = \frac{1}{2}(b_m - \sum_{n=2}^{m-1} c_n\sum_{\sum_{l=1}^n k_l = m}\prod_{l=1}^n c_{k_l})$$

But proving $f(z)$ is analytic, that is $c_n = \mathcal{O}(R^n)$ is not so easy.

Can you help proving it, or find some conditions for $f(z)$ to be analytic ? (in particular when $g(z) = \sin(z)$)

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reuns
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  • the problem is that with $f(z) = \frac{ z}{1-R z}$ I get $f(f(z)) = \frac{\frac{ z}{1-R z}}{1-R \frac{ z}{1-R z}} = \frac{ z}{1-2Rz}$ i.e. the radius of convergence is divided by $2$, so the induction "$|c_n| < R^n$ for $n < m \implies |c_m| < R^m$" doesn't work – reuns Sep 06 '16 at 07:00
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    I.N.Baker has done such a proof for the half-iterate of the $\exp(x)-1$ function. It is way over my head, but if you have a good math education you might get his arguments. A good entry point is perhaps the 1967 article "Non-embeddable functions with a fixpoint of multiplier 1" which starts with exactly your example and question. – Gottfried Helms Sep 06 '16 at 07:32
  • For $f(f(z)) = \sin z$: Iterative roots of sine, which has links for further reading (e.g. from Baker). – Martin R Sep 06 '16 at 07:51
  • The most relevant answer for the question of radius of convergence being zero or not is perhaps already given at http://mathoverflow.net/a/46791/7710 – Gottfried Helms Sep 06 '16 at 08:17

2 Answers2

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I see Gottfried has answered in various places. Good.

There is a real-valued $C^\infty$ solution, and this is $C^\omega$ except at the origin. It is in Gevrey class at the origin. To be specific, the solution can be extended to a holomorphic solution in a funny diamond shaped open region with two vertices at real $0$ and $ \pi. $ This should be thought of a the intersection of a sector between rays beginning at the origin, with a sector facing backwards ending at $\pi.$ The solution simply cannot be extended around the origin; this is quite visible in Ecalle's method, where a logarithm is an essential part of things.

The best books with relevant material are Milnor Dynamics in One Complex Variable and Kuczma, Choczewski, and Ger Iterative Functional Equations.

Will Jagy
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  • and can you relate Ecalle's method (that I didn't understand yet) with my recursive formula (at the origin, once $c_1= 1$, my coefs are unique) and prove they are not $\mathcal{O}(R^n)$ ? – reuns Sep 06 '16 at 18:40
  • and I think Gottfried or someone suggested my formal series can be regularized to yield a $C^\infty$ function (how ?) – reuns Sep 06 '16 at 18:41
  • @user1952009 I wouldn't know how to relate your methods with Ecalle. I recall that the KCG book proves uniqueness among $C^1$ fractional iterates, so they are presumably the same thing. – Will Jagy Sep 06 '16 at 19:13
  • @user1952009 : your formula looks like my implementation for the square-root of triangular carleman matrices, and thus I think your coefficients/ your solution for the formal powerseries is just the same which I produce in the MO-thread. I cannot/could not yet convert that Carlemanmatrix-approach(which is always my standard) into that of Ecalle, so I cannot say, they are formally equal. But I've done a numerical comparision of the Ecalle-Abel and the Carlemanmatrix-solution down to 40 digits precision, so the two methods are possibly even formally exchangeable - but no proof of that! – Gottfried Helms Sep 06 '16 at 20:22
  • @user1952009 : about the "regularization" of the divergent series: similarly like Cesaro- or Euler-summation there is a bit more general method:"Noerlund-summation" which can be configured to sum alternating series of the geometric and even hypergeometric types, where your powerseries (I assume it's really the matrix-squareroot-solution) seems to be of the latter type (I've now the first 1024 coefficients of the powerseries trying to extrapolate a useful formula for their growth). The summation method is not difficult (matrix-multiplication). If of interest I could put it into an "answer"-box – Gottfried Helms Sep 06 '16 at 20:29
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I've just looked at the 1967 article "Non-Embeddable Functions with a Fixpoint of Multiplier 1" of I.N.Baker (Math. Zeitschr. 99, 377-- 384 (1967)) and it seems, that it answers -at least to a big part- your question. The keyword "Embeddable" means here that such function can (or cannot, when "not embeddable") have an analytic/continuous iteration, which is analoguous to your question, whether/when fractional iterates have power series with nonzero convergence radius. I've just taken a screen shot of the second page of the article, perhaps this is useful:
image

Gottfried Helms
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