For two polynomials $a(X) = \sum_{i=0}^{N-1}a_i X^i$ and $b(X) = \sum_{i=0}^{N-1}b _i X^i$, its product is $c(X) = \sum_{i=0}^{2N-2} (\sum_{j=0}^{i}a_j b_{i-j})X^i$.
With distinct 2N-1 points $(v_0, v_1, \ldots, v_{2N-2})$ the two polynomials can be re-expressed as vectors in point-value representation:
$\vec{a} = (a(v_0), a(v_1), \ldots, a(v_{2N-2}))$
$\vec{b} = (b(v_0), b(v_1), \ldots, b(v_{2N-2}))$.
And, the point-value presentation of the corresponding product is as
$\vec{c} = (a(v_0)b(v_0), a(v_1)b(v_1), \ldots, a(v_{2N-2})b(v_{2N-2}))$.
The point-value representation can be converted back to the coefficient representation via Lagrange's interpolation.
How can I prove that point-valued hadamard product enables polynomial multiplication in coefficient representation (convolution)?
I tried to do use lagranges interpolation formula. But I could not realize this works.