A false interpolation paradigm (?); a reconsideration

[Gottfried]

The interpolation you describe is what Ansus also uses as Newton/Lagrange interpolation. (The interpolation polynomial is uniquely determined whether you use Newton or Lagrange formula or calculate it directly as solving the matrix equation like you do.)

I showed that this is interpolation (which only works for b<=e^(1/e)) is equivalent to the regular iteration at the lower (attracting) fixed point.

yes, and the wider our "unification-" project reaches, the more methods will be adressed by this consideration...

In so far the interpolation is "correct".

Yes, may be my wording is not explanative enough. I didn't want to say it were not correct in so far - the "false" logarithm series is also "so far correct" in its own set of conditions/requirements, it is completely legitimate in that setting: just in terms of an interpolation of some function in x. (we had this argument often: interpolation is only unique modulo some 1-periodic function)

But only in the wider view that set of conditions/requirements is not useful for the original intention, which was laid on the conceptually wider idea of a *logarithm*, namely to make log(a*b) = log(a) + log(b) and something more.
As we get by regular iteration some correct interpolated results, they satisfy many conditions correctly - but as I also understood our search for uniqueness, there is something more, for which the described method of interpolation is not sufficient. (I hoped Dmitriis integral approach would answer to such wider conditions, perhaps you both get it working soon)

There are two problems that occur in your description with the logarithm.
1. the logarithm has no powerseries at 0 (perhaps better try to interpolate log(x+1))

Ok, I also thought about this; I'll try this later

2. the interpolation method seems only to converge if \( f^{[n]} \) tends to a limit. Which the logarithm does not satisfy.

Hmmm. How far can this argument be related to that interpolation-method in more generality?
understand me correctly: I'm not on a search for a new interpolation-method for the logarithm, but I want to make things explicite down to the last "quark" . Then i am hoping to find a key, how such a method could be adapted/corrected/transformed to lead to the "correct" result ("correct" here in the sense for the wider scene) for instance in the easier example (with logarithm) and then see, what would such a correction/ transformation do with our regular iteration.

Hmm. I hope I did not create more obfuscation than clearing...

[update] ... and that I'm not only on some spleen idea... [/update]

Gottfried