12/11/2014, 01:22 PM
When discussing fake function theory we came across the fake sqrt.
the fake abs function is then fake_abs(x) = fake_sqrt(x^2).
Clearly when we want a real-entire function f(x) to satisfy f(x) = f(-x) then we consider f(fake_abs(x)).
But what if we have a real-entire function f(x) and we want to remove the property f(x) = f(-x).
We can do many things like : g(x) = f(x) + exp(-x). (*)
But lets consider the context of fake functions :
Find g such that
f(x) = g(fake_abs(x))
=>
g(x) = f(fake_abs^[-1](x))
g(x) = f( sqrt ( inv_fake_sqrt(x) ) )
Lets call inv_fake_sqrt(x) := fakesquare(x).
g(x) = f ( sqrt ( fakesquare(x) ) )
Now since f(x) = f(-x) we have that F(x) = f(sqrt(x)) is also an entire function.
g(x) = F( fakesquare(x) )
Now if we want g(x) to be entire then F ( fakesquare(x) ) needs to be entire.
Since fakesquare is a multivalued function ( an inverse of an entire ) its not entire.
SO when is F( fakesquare(x) ) entire ?
And how does that look like ?
Also of interest ( when its not entire ) :
fake ( F ( fakesquare(x) ) )
From (*) one then also wonders about
F( fakesquare(x) ) - f(x)
and how that looks like.
I have some ideas and guesses but no evidence or plots.
Seems like chapter 2 in fake function theory.
The analogue questions exist for exp(x) instead of x^2 ;
removing the periodic property.
regards
tommy1729
the fake abs function is then fake_abs(x) = fake_sqrt(x^2).
Clearly when we want a real-entire function f(x) to satisfy f(x) = f(-x) then we consider f(fake_abs(x)).
But what if we have a real-entire function f(x) and we want to remove the property f(x) = f(-x).
We can do many things like : g(x) = f(x) + exp(-x). (*)
But lets consider the context of fake functions :
Find g such that
f(x) = g(fake_abs(x))
=>
g(x) = f(fake_abs^[-1](x))
g(x) = f( sqrt ( inv_fake_sqrt(x) ) )
Lets call inv_fake_sqrt(x) := fakesquare(x).
g(x) = f ( sqrt ( fakesquare(x) ) )
Now since f(x) = f(-x) we have that F(x) = f(sqrt(x)) is also an entire function.
g(x) = F( fakesquare(x) )
Now if we want g(x) to be entire then F ( fakesquare(x) ) needs to be entire.
Since fakesquare is a multivalued function ( an inverse of an entire ) its not entire.
SO when is F( fakesquare(x) ) entire ?
And how does that look like ?
Also of interest ( when its not entire ) :
fake ( F ( fakesquare(x) ) )
From (*) one then also wonders about
F( fakesquare(x) ) - f(x)
and how that looks like.
I have some ideas and guesses but no evidence or plots.
Seems like chapter 2 in fake function theory.
The analogue questions exist for exp(x) instead of x^2 ;
removing the periodic property.
regards
tommy1729