(07/15/2022, 11:22 AM)Daniel Wrote: Gottfried, do you have any feel for which techniques discussed and developed here give equivalent results, at least for the sin function?

Well, Daniel, I never did this conclusively. But I had some identities based on analysis of the involved series. I've done what came up sometimes and put it on my webspace; partially with -perhaps- ridiculous low level of mathematical rigorosity (and I knew that ... so I didn't make big trumpet about it).

That analyses ran mostly towards comparing some method with the diagonalization, which over the years seemed to me the most complete formalism to express our iterations.

Keywords may be: my found interpolation ansatz "exponential polynomial interpolation" how I christened it, finding it is identical with the diagonalization (this includes the ansatz using q-binomials tried by Vladimir Reshetnikov in MO which we talked about recently), I found that the use of Newton-composition on a series of integer iterates finally is convertible with diagonalization, only the order of approximation is changed and so convergence issues might better or worse be matched by this or that method.

I tried to find compatibility of Andrew's ansatz (=Peter Walker's 2nd method) with diagonalization but couldn't make it match, instead it seemed that the evaluations via Andrew's ansatz and the diagonalization run into systematically different approximations, but could not make out the exact point where the difference occurs - and whether there is perhaps only one missing link, only one missing term in some series. Related with that ansatz of Andrew was one of mine using composition of the series-of-integer-iterates (I called this "AIS" - from "alternating iteration series", because if it is alternating it is easier to apply summation procedures at it than when the iteration series does not use alternating signs). (I don't have the workouts about Andrew's method and my AIS experimenting on the webspace, but I think the main aspects are here available in forum-contributions).

Very basic ideas on diagonalization (which came out to be finally equivalent to the Schroeder mechanism) are in the small essay "continuous functional iteration" and the most complete description of the set of coefficients in the diagonalization is surely "Eigensystem decomposition"

Just for the moment, possibly I can say more later/tomorrow -

Gottfried

Gottfried Helms, Kassel