Hi everyone!
Consider the sequence of Laurent polynomials (i.e. polynomials, in which negative degrees are allowed)
and another sequence of Laurent polynomials
As it turns out, these two sequences have the same linear span. What it practically means, is that every element of $$$B_k$$$ may be represented as a finite linear combination of the elements of $$$A_k$$$, and vice versa. It is somewhat trivial in one direction:
With some special consideration for even $$$k$$$, as the middle coefficient should be additionally divided by $$$2$$$. Now, the inverse transform is much trickier. To find it, consider the substitution $$$x=e^{i \theta}$$$, then you will notice that