Inverse function and fourier series

[zetafunction]

let be the Fourier expansion of the function

$$f(x) = \sum_{m=-\infty}^{m=\infty}c_{m} exp(imx)$$

valid on the interval (-1,1) , from this can we obtain the inverse function

$$f^{-1} (x)$$ by reflection of the Fourier series through the line y=x ??

 

[JJacquelin]

can we obtain the inverse function f^(-1)(x) by reflection of the Fourier series through the line y=x ??

Since exp(imx)=cos(mx)+i sin(mx), the function f(x) is complexe z=f(x)
Formally, the inverse function would not be f^(-1)(x) but rather x=f^(-1)(z)
What do you mean in writing "by reflection through the line y=x " ?
This would have a sens in case of reals y=f(x) and x=f^(-1)(y). Presently it is not a real y, but a complex z instead of y.
What do you reflect through the line y=x ? Is it the curve representing y(x)= real part of z as a function of x ?, or is it the curve representing y(x)= imaginary part of z as a function of x ? or is it the curve representing y(x)= module of z as a function of x ? or is it the curve representing y(x)= argument of z as a function of x ?

 

[zetafunction]

assume function f(x) is real, then the Fourier series will include sines and cosines only .. but no exponential quantities.

 

[JJacquelin]

assume function f(x) is real, then the Fourier series will include sines and cosines only .. but no exponential quantities

Sure ! if f(x) is assumed to be real the general terms of the Fourier series will be :
a_mcos(mx)+b_msin(mx) which is real.
But in the first question this was not the case : The general term was c_mexp(i m x) = c_m( cos(mx) + i sin(mx) ) which is not real.
So the question was raised in the general case of f(x) complex, even if the coefficients c_m are complex (and/or real).

 

[chiro]

let be the Fourier expansion of the function

$$f(x) = \sum_{m=-\infty}^{m=\infty}c_{m} exp(imx)$$

valid on the interval (-1,1) , from this can we obtain the inverse function

$$f^{-1} (x)$$ by reflection of the Fourier series through the line y=x ??

I'm interested in this as well if anyone has any ideas.

 

[JJacquelin]

Well, the problem cannot be raised in terms of reflexion through the line y=x on a real space.
Seriously, expressing the reciprocal of a Fourier series is a very difficult problem, still open :
http://www.jstor.org/pss/2031811
http://www.jstor.org/pss/2034097