02/17/2009, 09:25 PM
As we know a real analytic Abel function \( \alpha(x)=\sum_{n=0}^\infty \alpha_n x^n \) of \( f \), i.e. a function satisfying \( \alpha(f(x))=\alpha(x)+1 \)
can be obtained by solving the infinite equation system:
\( \sum_{n=0}^\infty \alpha_n {f^n}_m = \alpha_m + \delta_{m,0} \), \( m\ge 0 \)
where \( f^n \) denotes the \( n \)-th power (not iteration) of \( f \). \( \alpha_0 \) is undetermined, as it is for Abel functions (if \( \alpha \) is a solution then \( \alpha(x)+c \) is also a solution).
This equation system is equivalent to:
\( \sum_{n=1}^\infty \alpha_n ({f^n}_m-\delta_{m,n}) = \delta_{m,0} \), \( m\ge 0 \).
Now Erhard Schmidt [1] developed 1908 a method to solve infinite equation systems, especially a method to obtain a unique solution such that \( \sum_{n=0}^\infty |\alpha_n|^2 \) is minimal. By an article [2] which I couldnt not get hold of yet this method may yield the same solution as the here already discussed "natural"/Walker/Robbins method of taking the limit of the truncated systems solution. Just wanted to throw this into the uniqueness discussion.
[1] Erhart Schmidt. Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten, Palermo Rend. 25 (190
, 53-77
[2] B. V. Krukovskii. On the relation between some formulas of the determinantal and nondeterminantal theories of linear equations with an infinite number of unknowns (Russian), Kiev. Avtomobil.-Doroz. Inst. Trudy 2 (1955), 176-188.
can be obtained by solving the infinite equation system:
\( \sum_{n=0}^\infty \alpha_n {f^n}_m = \alpha_m + \delta_{m,0} \), \( m\ge 0 \)
where \( f^n \) denotes the \( n \)-th power (not iteration) of \( f \). \( \alpha_0 \) is undetermined, as it is for Abel functions (if \( \alpha \) is a solution then \( \alpha(x)+c \) is also a solution).
This equation system is equivalent to:
\( \sum_{n=1}^\infty \alpha_n ({f^n}_m-\delta_{m,n}) = \delta_{m,0} \), \( m\ge 0 \).
Now Erhard Schmidt [1] developed 1908 a method to solve infinite equation systems, especially a method to obtain a unique solution such that \( \sum_{n=0}^\infty |\alpha_n|^2 \) is minimal. By an article [2] which I couldnt not get hold of yet this method may yield the same solution as the here already discussed "natural"/Walker/Robbins method of taking the limit of the truncated systems solution. Just wanted to throw this into the uniqueness discussion.
[1] Erhart Schmidt. Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten, Palermo Rend. 25 (190
, 53-77[2] B. V. Krukovskii. On the relation between some formulas of the determinantal and nondeterminantal theories of linear equations with an infinite number of unknowns (Russian), Kiev. Avtomobil.-Doroz. Inst. Trudy 2 (1955), 176-188.