I have a periodic term $V(x) = \sum_K \exp(iKx) V_K$ where $K =2\pi n/a$ where $a$ is the periodicity of the term and $n =0,1,2,3....$
Now I want to find the Fourier coefficient $V_K$ corresponding to a particular $K$. Suppose I have a vector for $V(x)$ having $10000$ points for $x = 0,0.01a,0.02a,...a,1.01a,....2a....100a$ such that the size of my lattice is $100a$.
FFT on this vector gives $10000$ Fourier coefficients. The $K$ values corresponding to these Fourier coefficients are $2\pi n/(10000*0.01a)$ with $n=0,1,2,3,...9999$. But my $K$ had the form $2\pi n/a$ due to the periodicity of the lattice.
What am I missing ?