- Premise: I was attempting to use Discrete Time Fourier Transforms as a tool to solve PDEs for the first time and for my first example I decided to try it on a simple Boundary Value Problem on a boundary from $x=[0,1]$. However the Discrete Fourier Transform did not converge to the correct Fourier Transform needed for obtaining the solution until I used an odd extension.
Setup for the problem is below:
$$\cases{u(x_i)_{xx} = f(x_i), & $x\in [0,1], i\in[0,N]$;\\\ u(0) = 1, u(1) = 2 \\f(x_i) = 2}$$ It's a fairly simple BVP with solution $u(x) = x^2+1$
Extension: $$g(x_i) = \cases{f(x_i) = f(x_i), & $x\in (0,1], i\in(N,2N]$;\\\ f(x_i) = -f(x_i), & $x\in [-1,0], i\in[0,N]$}$$
When using Fourier Transforms on $f(x_i)$ to obtain $u(x_i)$ the inverse Fourier integrals did not converge to the correct solution for $u(x_i)$, however they did when I added an odd extension to $f(x_i)$ to make $g(x_i)$ and applied the Discrete Fourier Transform to that instead.
- Question: I saw this question answered here to some extent, but did not understand how this worked for a Discrete Time Fourier Transform or what cases it works in. So, before applying DFTs to functions/signals, is it always appropriate to modify them by applying Odd Extensions? Or does it vary from case-to-case?
SIDENOTE: I can include the workings for Discrete Fourier Trasnforming everything and deriving this all out to solve for $u(x)$, but I'm pretty sure this question is more conceptual in nature so I'll leave the workings out for now to make the question less cluttered unless someone feels it necessary to see everything.