bo198214 Wrote:You probably mean
B * W(B)[col] = W(B)[col] * d[col]
but thats just the matrix version of the Schroeder equation \( \sigma(\exp_b(x))=c\sigma(x) \).
If you find a solution \( \sigma \) to the Schroeder equation you make the Carleman/column matrix \( W(B) \) out of it and you have a solution of this infinite matrix equation and vice versa if you have a solution W(B) to the above matrix equation that is the Carleman/column matrix of a powerseries \( \sigma \) then this power series is a solution to the Schroeder equation. Nothing new is gained by this consideration.
I'm trying to understand the problem of uniqueness now. Since by the eigensystem I do not only have one equation (imaging the meaning of fixpoints) for the first row by
W^-1 [,0] * Bs = d[0,0] * W^-1[,0] = 1*W^-1[,0]
whose series-expression may not be a non-uniquely determination for f(x) = s^x (or another way round?)
(Henryk suggested some examples, for instance a mixture using coefficients of an overlaid sine-function, if I understood this correctly).
I have infinitely many equations according to each row of W^-1. Say u = log(t) wheren t^(1/t) = b = base I have also
W^-1 [,1] * Bs = u^1*W^-1[,1]
W^-1 [,2] * Bs = u^2*W^-1[,2]
...
(where the second row happens to express the series for the first derivative w.r. to the topexponent x if I see it right)
and so on.
The same sine-overlay should not be sufficient to satisfy all these further equations.
But this is merely speculation here, since I did not yet understand, in what the uniqueness-problem comes into play.
Gottfried
Gottfried Helms, Kassel