Our first example is that of a multiplication between two polynomials, which we will rewrite in terms of the mathematics of convolution. In the final stage, since the problem here is a discrete one, we will recast the final expression in terms of a matrix-vector multiplication, where the matrix is a so-called Toeplitz matrix .
Let us look a the following polynomials to second and third order, respectively:
$$ p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, $$and
$$ s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. $$The polynomial multiplication gives us a new polynomial of degree \( 5 \)
$$ z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. $$