Matrix-method: compare use of different fixpoints

[Gottfried]

I want your acknowledgement on this.

Well, you say it in the next sentence: let truncation aside. In all my considerations ...

Is this a confirmation?

I can only restate: in no situation I assumed the matrices as truncated by principle - all my considerations assume the unavoidable truncations in praxi as giving approximations in numerical evaluation as plcaeholder for the basically infinite matrices. I never discussed a finite (truncated) matrix being more than such an approximation for determination of any intermediate result.
For instance: if the carleman-matrix is thought of finite size, then I don't see any relation between carleman matrices and any of my matrices.
I don't know, whether this satisfies your request (if not, then please give even more explanation of what is the core of your question, I must then being unable to understand the relevant implications correctly)

Gottfried though I am not 100% clear about your my-oh-my method, it seems as if it does nothing more than to find the regular iteration at a fixed point, we discussed something similar already here.

That may all be - and whether it is "nothing more" or not: if it is the regular iteration, then fine; if not, then fine again.

The regular iteration of a function \( f \) with fixed point at 0 is given by
\( f^{\circ t} = \sigma^{-1}\circ \mu_{c^t} \circ \sigma \) with \( c=f'(0) \), \( \mu_d(x)=d\cdot x \) and \( \sigma \) being the principal Schroeder function.
If the fixed point is not at 0 but is at \( a \) then \( f=\tau_{-a}\circ \exp_b \circ \tau_a \) is a function with fixed point at 0 and \( f'(0)=c=\ln(a) \), where \( \tau_d(x)=d+x \). Putting this into the first equation we get:
\( \exp_b^{\circ t} = \tau_a\circ\sigma^{-1}\circ\mu_{{\ln(a)}^t}\circ\sigma\circ \tau_{-a} \).

But if we translate this into matrix notation by simply replacing \( \circ \) by the matrix multiplication and replacing each function by the corresponding Bell matrix (which is the transposed Carleman matrix) then we see a diagonalization of \( B_b \) because the Bell matrix (and also the Carleman matrix) of \( \mu_{\ln(a)}(x)=\ln(a)x \) is just your diagonal matrix \( {_dV}(\ln(a)) \)!

Yes, this seems so - but did we not already state the identity of the matrix B (or Bs) with the Bell/Carleman-transposes? I thought, that this had settled the question already? I was very happy, when you pointed out the relation in one of your previous posts - I couldn't have done it due to my lack of understanding of those concepts (described in elaborated articles, more than I could follow in detail).

So it is nothing new, that we can diagonalize the untruncated \( B_b \) for each fixed point \( a \) with the diagonal matrix \( {_dV}(\ln(a)) \). As result we only get the plain old regular iteration at a fixed point.

Hmm, for me this is an *achievement*, so my bottom up approach just out of the sandbox is then decoded into the terminology of "Schroeder-equation" and "regular iteration" - so: good!

If from there some shortcomings of the method are already *known* then I would like to know them, too.

Hmm, I've no more idea at this moment. I'll reread your post later this evening, perhaps I'm missing some point.

Gottfried