Let $F_2$ be the set of all the functions from the finite field $GF(2^n)$ of $2^n$ elements to $GF(2)$. I am reading a textbook that proves that the elements of $F_2$ can be represented by polynomials; the authors use the Lagrange Interpolation Formula.
They say:
Any function $f\in F_2$ using Lagrange interpolation and noticing that $x^{2^n}=x $ for $x\in GF(2^n)$ can be represented as a polynomial of degree $\leq 2^n-1$. In other words, we may define the (discrete) Fourier transform for functions in $F_2$ in terms of Lagrange interpolation.
Definition: For $f\in F_2$ the discrete Fourier Transform of $f$ is defined to be $ A_k=\sum_{x\in {GF(2^n)}^*} f(x)x^{-k},\quad k=1,\dots,\,2^n-1,\; A_0=f(0) $
They also note that $A_{2^{n}-1}=\sum_{x\in {GF(2^n)}^*}f(x)$. So far so good. But then they simply state that:
The inverse of the formula [above] is given as follows: $f(x)=\sum_{k=0}^{2^n-1}A_k x^k, x\in GF(x^n)^*$
Then they go on and say that this means that $F_2$ consists of polynomials.
Could somebody please explain or give some hints about how they got that inverse above? In particular, how exactly was the Lagrange interpolation formula used?