02/17/2009, 09:25 PM

As we know a real analytic Abel function of , i.e. a function satisfying

can be obtained by solving the infinite equation system:

,

where denotes the -th power (not iteration) of . is undetermined, as it is for Abel functions (if is a solution then is also a solution).

This equation system is equivalent to:

, .

Now Erhard Schmidt [1] developed 1908 a method to solve infinite equation systems, especially a method to obtain a unique solution such that 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:

,

where denotes the -th power (not iteration) of . is undetermined, as it is for Abel functions (if is a solution then is also a solution).

This equation system is equivalent to:

, .

Now Erhard Schmidt [1] developed 1908 a method to solve infinite equation systems, especially a method to obtain a unique solution such that 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.