This question is within the context of developing the discrete Fourier interpolation. We begin with an interval $[a,b]$ with a uniform partition $a = x_0 < ··· < x_{N−1} < b$, where each $x_i = x_{i−1} + ∆x = x_0 + i∆x$. Taking the usual variable change, we may suppose that $a = 0, b = 2π = N∆x$ and $x_i = i∆x$. For two complex valued functions $f$ and $g$ defined on
{${x_0,x_1,...,x_{N−1}}$}, we set $σ(f,g) = \sum_{i=0}^{N-1}f(x_i)\overline{g(x_i)}$.
Prove that if $g(x_j) = \sum_{k=-M_0}^{M_1 - 1} c_ke^{\textbf{i}kj∆x}$, then for each $j, g(x_j) = f(x_j)$.
(Hint: Expand
$\sum_{k=-M_0}^{M_1 - 1} c_ke^{\textbf{i}kj∆x}$
$=\frac{1}{N} \sum_{k=-M_0}^{M_1 - 1} σ(f,e^{\textbf{i}kx}) e^{\textbf{i}kj∆x}$
$=\frac{1}{N} \sum_{k=-M_0}^{M_1 - 1} (\sum_{l=0}^{N - 1} f(x_l) e^{\textbf{-i}kx_l}) e^{\textbf{i}kj∆x}$
and keep in mind that $e^{\textbf{i}kj∆x} = e^{\textbf{i}kx_j} = e^{\textbf{i}jx_k}$)