1

Lately I have been investigating numerical methods to approximate fractional composition roots of functions. My intuition tells me that power series expansions will not be very suitable in approximating fractional functions in the sense

$$(f^{\circ 1/k})^{\circ k}(t) = f(t)$$

For example

$g(t) = \sin^{\circ 1/16}(t)$ would be any function satsifying $g^{\circ 16}(t) = \sin(t)$. In other words $\underset{16 \hspace{0.166cm}\text{times}}{\underbrace{g(g(\cdots g(g(t}}))\cdots)) =\sin(t) $

Any well behaved function which is such a solution $t\to g(t)$ will be extremely smooth and rather close to $f(t) = t$ so that a normal power series expansion will be unsuitable. The monomials are simply diverging way too fast.

I have been hypothesizing that families with linear combinations of $f(t) = t$ and in the exponential family with negative exponents could be meaningful. Or maybe logarithmic functions.

But I can't think of a theoretical approach to find reasonable families of functions.

I am curious of where to find sources for how to approach this.

Answers are of course welcome as well.

mathreadler
  • 25,824
  • 1
    Perhaps you're interested in my very basic discussion of that iteration thingy... However, I make the course towards powerseries, and where I arrive at an example showing the iteration if the sine-function. See this https://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf . At the siteindex https://go.helms-net.de/math/tetdocs/index.htm there are some more materials, for instance iteration of $ f(x)= 1/(1+x)$ (where I show some way without powerseries) and something more. – Gottfried Helms Oct 30 '22 at 08:42
  • 1
    @GottfriedHelms I do like Carleman and Bell matrices in analyzing functional composition, however I am not so optimistic when it comes to using power series expansions for representing fractional functions. I made some experiments some time back and found polynomial bases to be too fast-changing to fit well to functions which are close to the identity function. – mathreadler Oct 30 '22 at 17:21
  • 1
    I see. Maybe because of my frequent use of procedures for summation of divergent series I became too unaware of this specific type of problem. I'll give it a bit more attention in the future... – Gottfried Helms Oct 30 '22 at 18:42
  • @GottfriedHelms You don't need to haste it with lack of rigor if you don't want to. I developed a couple of techniques on my own. – mathreadler Nov 01 '22 at 11:35
  • 1
    Well, it's alright. I'd had doubts myself concerning the appropriateness/optimality of powerseries, and thought of using Fourierseries or similar things, but have zero knowledge about that type of operations. Just recently found one Q. here in MSE ("Diagonalization of Besselmatrix") - Ok, I've to leave for today... – Gottfried Helms Nov 01 '22 at 12:08
  • @GottfriedHelms I suppose that you mean https://math.stackexchange.com/questions/4297398/fourier-series-of-iterated-sin-diagonalization-of-an-infinite-matrix-of-bessel Hmm, that was curious. Thank you. I wonder if one can do the same for other families as well. Will it give the same result or which factors will be affeting? – mathreadler Nov 01 '22 at 12:21
  • I don't know about the other families; I'm totally illiterate with fourieranalysis. First idea is of course to apply to $\cos()$ - but I've no idea what the Besselmatrix has to look like and the series composition as well. The "obvious" insertions was not successful... ' – Gottfried Helms Nov 01 '22 at 18:14
  • @GottfriedHelms I've done some Fourier Analysis, but I guess you can say Bessel functions are a "black hole" for me.. – mathreadler Nov 02 '22 at 09:34

0 Answers0