Nautilus
04-10-2005, 10:51 PM
OK. here is some work on Sunday evening, which is a real fun at least for some people.
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my prior would be along Koshi-Adamara (russian transcription) formula, which is the radius (R) of a circle where the Tailor expansion holds is determined by
suplim (c_n)^(1/n)=1/R, where
c_n is an n derivative.
if derivatives grow at most exponentially fast, then the expansion holds for all x and u, right?
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yep, using sterling's formula for n! it is easy to give an answer in terms of bounds on derivatives
if derivatives are bounded after some order, then the expansion must hold for all x and u
fun fun fun
At 09:20 PM 4/10/2005 -0400, you wrote:
well, with a fear of making a blank statement let me indulge myself with a line of reasoning (borrowed from Complex analysis). In complex plane Taylor expansion is valid within a circle that has a critical point on the boundary. (A good example is 1/(1+x^2), point i is a critical one so the convergence radius is one (if you start from 0)). Now we know that any analytical function can be extended so the above logic should hold (that is if we inclose a set in R into complex plane and extend the function into it, then it must be that the extended function does not have critical points within the area of interest). I suspect that it is also possible to give an answer in terms of norms of derivatives.
At 09:06 PM 4/10/2005 -0400, you wrote:
I think real analyticity requires u to be "sufficiently small"
At 09:01 PM 4/10/2005 -0400, you wrote:
As far as I remember a function has to be analytic (which is slightly different from infinite differentiability). I don't have a right book at hand to make this statement precise and rigorous now. I can look for it tomorrow.
At 08:51 PM 4/10/2005 -0400, you wrote:
Under what conditions does a function coincide with its infinite Taylor expansion at all points, i.e when does
f(x + u) = f(x) + f'(x) (u) + ...
hold for all x in R and all u in R (or R replaced by a compact set K)?
Is this a stringent condition for infinitely differentiable functions defined over compact set?
---------------------
my prior would be along Koshi-Adamara (russian transcription) formula, which is the radius (R) of a circle where the Tailor expansion holds is determined by
suplim (c_n)^(1/n)=1/R, where
c_n is an n derivative.
if derivatives grow at most exponentially fast, then the expansion holds for all x and u, right?
-------------------
yep, using sterling's formula for n! it is easy to give an answer in terms of bounds on derivatives
if derivatives are bounded after some order, then the expansion must hold for all x and u
fun fun fun
At 09:20 PM 4/10/2005 -0400, you wrote:
well, with a fear of making a blank statement let me indulge myself with a line of reasoning (borrowed from Complex analysis). In complex plane Taylor expansion is valid within a circle that has a critical point on the boundary. (A good example is 1/(1+x^2), point i is a critical one so the convergence radius is one (if you start from 0)). Now we know that any analytical function can be extended so the above logic should hold (that is if we inclose a set in R into complex plane and extend the function into it, then it must be that the extended function does not have critical points within the area of interest). I suspect that it is also possible to give an answer in terms of norms of derivatives.
At 09:06 PM 4/10/2005 -0400, you wrote:
I think real analyticity requires u to be "sufficiently small"
At 09:01 PM 4/10/2005 -0400, you wrote:
As far as I remember a function has to be analytic (which is slightly different from infinite differentiability). I don't have a right book at hand to make this statement precise and rigorous now. I can look for it tomorrow.
At 08:51 PM 4/10/2005 -0400, you wrote:
Under what conditions does a function coincide with its infinite Taylor expansion at all points, i.e when does
f(x + u) = f(x) + f'(x) (u) + ...
hold for all x in R and all u in R (or R replaced by a compact set K)?
Is this a stringent condition for infinitely differentiable functions defined over compact set?