generalizing the problem of fractional analytic Ackermann functions

[Gottfried]

Giving us what I put forth earlier

the half superfunction of the half superfunction of f is the superfunction of f.

\( F^{\frac{1}{2}} \{ F^{\frac{1}{2}} \{ f \} \} (x) = F \{ f \} (x) \)

This definition cannot fall victim to the method of disproof Tommy gave forth because the transformation notation takes three arguments whereas the diamond notation only takes two.

Hmm, this looks then as if this is -in terms of Carleman-matrices- a continuation of the diagonalization. Assume we have alsready a diagonalization of our carleman-matrix for (decremented) exponentiation dxp°h(x) B as \( B = W * D * W^{-1} \), and for the h'th iteration using the h'th power of D \( B^h = W * D^h * W^{-1} \), then your logic seems to me the idea to diagonalize W and give it fractional powers: \( W = V * E * V^{-1} \), and \( W^g = V * E^g * V^{-1} \) such that we have \( B_g^h = (V *E^g*V^{-1} ) * D^h *( V * E^{-g}*V^{-1} ) \) where the *g* gives the "rate" of the superfunction.

We had one time a small discussion about this (I don't remember the thread, perhaps I can provide the reference later); I'd observed, that the three operations addition, multiplication, exponentiation could be listed by powers of W: \( W^{-1}, W^0 \text{ and } W^1 \) respectively . But I didn't proceed here because of some "unevenness" with this expression for addition. But well: if this meets your idea at all then why not try and find a better extrapolation/embedding now than that sketchy discussion which we didn't continue ...

Gottfried

[update] the link to the earlier thread: http://math.eretrandre.org/tetrationforu...hp?tid=364