When doing Von Neumann Stability Anaylsis Id how does one go

When doing Von Neumann Stability Anaylsis, I\'d how does one go from the recurrence relation for the error, to a finite Fourier series describing teh spatial vartion of the error?
From the article in the link below, how do they go from equation (2) to equation (3)?

https://en.wikipedia.org/wiki/Von_Neumann_stability_analysis

Solution

2nd equation represents the error in the form of recurrence relation,after analyzing both the error and numerical solution, they have both the same variation wrt time

Since the numerical solution can be modeled as infinite fourier series for interpretation in order to have single sigma to find the stability analysis

Since the error equation follows the same pattern as the numerical equation, we can model the same as well with the fourier series as represented by the equation (3)