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 Least squares Abel function bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 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. « Next Oldest | Next Newest »

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