06/18/2023, 11:08 PM
I wanted to point out a kind of reverse of fake function searching.
Suppose we want to find the fake function for f(x).
then we suspect a candidate
f(x) = sum f_n x^n.
Now we try the following :
1) estimate max_n : max f_n x^n.
To do that set
f_a x^a = f_(a+1) x^(a+1)
This a = g(x) is an indicatator for the growth rate and dominant terms of the taylor series.
Notice it depends on x.
Now compute another similar indicator , this time for the inferior terms of the taylor series :
2) estimate max_b : f_b x^b < 1
you might want to set f_n x^n = (x/h_n)^n for convenience.
it follows x around h_n is getting close.
b = h(x) is again an indicator of the growth rate and number of dominant terms we at least need for our taylor.
We now found the " start of the tail " of the taylor series.
3) Using a = g(x) and b = h(x) , we are now kinda ready to estimate
I = integral f_n x^n d n from 0 to infinity.
by comparing to the estimate
(f_c + f_d) x^c = min f(x)
what is the basis for fake function theory :
=> min f(x) / x^c = (f_c + f_d)
and compare that f_c , f_d , c , d , a , b and the integral I to f(x) ,
giving us a good idea of how good our original estimate was and a way to improve it.
If f_c/(f_c + f_d) is close to *constant* you know fake function estimate was quite decent.
Or if lim f_c/(f_c + f_d) is close to *constant* you know fake function estimate was quite decent asymptotically.
This is a sketch of the idea , ofcourse calculus tricks can and should be applied !
But that application is case specific , since we might be working with complicated functions.
Also I used integral but ofcourse any approximation of the sum that is good will do, in particular - if possible - actual sum formulas.
Iterating these estimate methods always results in a good estimate.
I wanted to prove that formally but it might take some time.
On the other hand I am not sure of the conjectures made so far are correct.
Im also thinking about what tetration methods are best for the fake functions of tetration.
I have ideas but nothing I am very convinced of yet, let alone a proof.
It might take me some time.
But I can say the used fixpoints matter.
regards
tommy1729
Suppose we want to find the fake function for f(x).
then we suspect a candidate
f(x) = sum f_n x^n.
Now we try the following :
1) estimate max_n : max f_n x^n.
To do that set
f_a x^a = f_(a+1) x^(a+1)
This a = g(x) is an indicatator for the growth rate and dominant terms of the taylor series.
Notice it depends on x.
Now compute another similar indicator , this time for the inferior terms of the taylor series :
2) estimate max_b : f_b x^b < 1
you might want to set f_n x^n = (x/h_n)^n for convenience.
it follows x around h_n is getting close.
b = h(x) is again an indicator of the growth rate and number of dominant terms we at least need for our taylor.
We now found the " start of the tail " of the taylor series.
3) Using a = g(x) and b = h(x) , we are now kinda ready to estimate
I = integral f_n x^n d n from 0 to infinity.
by comparing to the estimate
(f_c + f_d) x^c = min f(x)
what is the basis for fake function theory :
=> min f(x) / x^c = (f_c + f_d)
and compare that f_c , f_d , c , d , a , b and the integral I to f(x) ,
giving us a good idea of how good our original estimate was and a way to improve it.
If f_c/(f_c + f_d) is close to *constant* you know fake function estimate was quite decent.
Or if lim f_c/(f_c + f_d) is close to *constant* you know fake function estimate was quite decent asymptotically.
This is a sketch of the idea , ofcourse calculus tricks can and should be applied !
But that application is case specific , since we might be working with complicated functions.
Also I used integral but ofcourse any approximation of the sum that is good will do, in particular - if possible - actual sum formulas.
Iterating these estimate methods always results in a good estimate.
I wanted to prove that formally but it might take some time.
On the other hand I am not sure of the conjectures made so far are correct.
Im also thinking about what tetration methods are best for the fake functions of tetration.
I have ideas but nothing I am very convinced of yet, let alone a proof.
It might take me some time.
But I can say the used fixpoints matter.
regards
tommy1729
