It may be a bit bizarre that I ask this question two or three years after I learned the use of the schroederfunction and the recentering of the powerseries for the purpose of fractional iteration...
Well the question surfaces just after looking at my sketchpad, where I've written the recentering-formalism for the powerseries and the schroeder-function together. We have for some function f(x), recentered at fixpoint t the iterable decomposition into Schroederfunctions, depending on the iteration-height h and the eigenvalue u:
\( f^{[h]}(t) = t+ \sigma^{-1} ( u^h * \sigma (x-t) ) \)
When I look at this I ask how it was justified to use the schroeder-functions with only partly recentering; shouldn't we expect:
\( f^{[h]}(t) = t+ \sigma^{-1} \left( u^h * \left(t+ \sigma (x-t) \right) -t \right) = t+ \sigma^{-1} \left( t (u^h-1)+ u^h \sigma (x-t) \right) \)
Always thinking in terms of the matrix-notations this seems more naturally to me (it is the similarity transformation). Well I've tried a quick check what would happen if I modified the standard computation this way and, as expected, I got salad. But why is this, that we allow to use the schroeder-function in context with a centered series where the centering is included only partly? (I have the article of Schroeder where he explains his method but don't remember (I'll look at it) that he was explicitely dealing with that recentering....)
Gottfried
Well the question surfaces just after looking at my sketchpad, where I've written the recentering-formalism for the powerseries and the schroeder-function together. We have for some function f(x), recentered at fixpoint t the iterable decomposition into Schroederfunctions, depending on the iteration-height h and the eigenvalue u:
\( f^{[h]}(t) = t+ \sigma^{-1} ( u^h * \sigma (x-t) ) \)
When I look at this I ask how it was justified to use the schroeder-functions with only partly recentering; shouldn't we expect:
\( f^{[h]}(t) = t+ \sigma^{-1} \left( u^h * \left(t+ \sigma (x-t) \right) -t \right) = t+ \sigma^{-1} \left( t (u^h-1)+ u^h \sigma (x-t) \right) \)
Always thinking in terms of the matrix-notations this seems more naturally to me (it is the similarity transformation). Well I've tried a quick check what would happen if I modified the standard computation this way and, as expected, I got salad. But why is this, that we allow to use the schroeder-function in context with a centered series where the centering is included only partly? (I have the article of Schroeder where he explains his method but don't remember (I'll look at it) that he was explicitely dealing with that recentering....)
Gottfried
Gottfried Helms, Kassel