06/30/2008, 09:23 PM
Using a matrix-expression this would be
t°h(x) = W^-1 sum k=0..inf sum j=0..k (-1)^j * binomial(k,j) *diag(1,u^j,u^2j,...) * W
sum j=0..k (-1)^j * binomial(k,j) *dV(u^j) = diag(u^j-1) *PPow(
sum k=0..inf
Hmm, let me try (maybe I didn't get this right yet).
\( f^{\circ t} = \sum_{n=0}^\infty \left(t\\n\right) \sum_{k=0}^n \left(n\\k\right) (-1)^{n-k} f^{\circ k} \)
is
\( f^{\circ t} = \sum_{n=0}^\infty \left(t\\n\right) c_n \)
which is just a binomial weighting of the coefficients c_n.
In my analyses I got the coefficients
\( f^{\circ t}(x) = \sum_{n=0} a_n x^n \)
so, for instance, the c_n for the half-iteration are
\( c_n = a_n / \left(t\\n\right) \)
The rate of growth of the a_n-coefficients for t=0.5 was asymptotically
\( a_n \sim~ u^{0.5} * \frac{u^{\frac{n^2-n}{2}}}{n!} * m_n \)
where m_n are also growing coefficients, if only the leading coefficient
of the polynomials at x^n are taken into account.
Now the quotient of two consecutive binomials
\( \frac{ \left(0.5\\n\right) }{\left( 0.5\\n+1\right) } \)
seem to approach -1, so the strong growth of about u^n^2/n!, or the
quotient of two consecutive coefficients of ~ u^2n/n seems to
dominate the characteristic of the c_n-coefficients.
A series with quotient of increasing absolute value u^(2n)/n, u>1 cannot
regularly be Euler-summed; maybe it can be summed with Borel-summation of
higher orders.
To be "not regularly" Euler-summable does not mean, we cannot have
an approximation of a certain degree; however the problem with
this is, that the partial sums may converge up to a certain
index n, from where it "begins to diverge" - and it is not yet
known to me, to what extent we can use the intermediate approximated
value - I'm investigating for verification of some experimental
summation-methods of the required power.
Hmm - i hope this is not more confusing than clarifying - I've my
head not really free today (have to prepare the final lesson tomorrow)
Gottfried
t°h(x) = W^-1 sum k=0..inf sum j=0..k (-1)^j * binomial(k,j) *diag(1,u^j,u^2j,...) * W
sum j=0..k (-1)^j * binomial(k,j) *dV(u^j) = diag(u^j-1) *PPow(
sum k=0..inf
bo198214 Wrote:\( f^{\circ t} = \sum_{n=0}^\infty \left(t\\n\right) \sum_{k=0}^n \left(n\\k\right) (-1)^{n-k} f^{\circ k} \)
Notes:
- Perhaps Gottfried can jump in to provide summability in the divergent case \( b>e^{1/e} \).
Hmm, let me try (maybe I didn't get this right yet).
\( f^{\circ t} = \sum_{n=0}^\infty \left(t\\n\right) \sum_{k=0}^n \left(n\\k\right) (-1)^{n-k} f^{\circ k} \)
is
\( f^{\circ t} = \sum_{n=0}^\infty \left(t\\n\right) c_n \)
which is just a binomial weighting of the coefficients c_n.
In my analyses I got the coefficients
\( f^{\circ t}(x) = \sum_{n=0} a_n x^n \)
so, for instance, the c_n for the half-iteration are
\( c_n = a_n / \left(t\\n\right) \)
The rate of growth of the a_n-coefficients for t=0.5 was asymptotically
\( a_n \sim~ u^{0.5} * \frac{u^{\frac{n^2-n}{2}}}{n!} * m_n \)
where m_n are also growing coefficients, if only the leading coefficient
of the polynomials at x^n are taken into account.
Now the quotient of two consecutive binomials
\( \frac{ \left(0.5\\n\right) }{\left( 0.5\\n+1\right) } \)
seem to approach -1, so the strong growth of about u^n^2/n!, or the
quotient of two consecutive coefficients of ~ u^2n/n seems to
dominate the characteristic of the c_n-coefficients.
A series with quotient of increasing absolute value u^(2n)/n, u>1 cannot
regularly be Euler-summed; maybe it can be summed with Borel-summation of
higher orders.
To be "not regularly" Euler-summable does not mean, we cannot have
an approximation of a certain degree; however the problem with
this is, that the partial sums may converge up to a certain
index n, from where it "begins to diverge" - and it is not yet
known to me, to what extent we can use the intermediate approximated
value - I'm investigating for verification of some experimental
summation-methods of the required power.
Hmm - i hope this is not more confusing than clarifying - I've my
head not really free today (have to prepare the final lesson tomorrow)
Gottfried
Gottfried Helms, Kassel