Efficiently Compute Inverse Fourier Transform

The following code was suggested in the very nice answer to this question here. My question about it is: If I was just to compute the derivative in Fourier space as done in this co

Solution 1:

Here one only needs to define the linear operator for the inverse transformation. Following the structure from the posted code it would be

expinv = np.outer( 1j * 2 * np.pi * xl, kl ) 
AInv = np.exp( expinv ) / np.sqrt( N )

The Fourier transform of the derivative was

dfF = np.dot( B, yl )

such that the derivative in real space would be

dfR = np.dot( AInv, dfF ) 

Eventually, this means that the "derivative"-operator is

np.dot( AInv, B )

which is for small N ( except for the edges ) a tridiagonal matrix with entries (-1,0,1), i.e. a classical symmetric numerical derivative. Increasing N it first changes to 1,-2,0,2,1 i.e. a higher order approximation of the derivative.

Eventually, one gets an alternating weighted derivative of type (..., d,-c, b,-a,0,a,-b, c,-d, ...)