Apologies if this question is vague/unclear (especially the title).
Given a vector $x=(x_0, \ldots, x_n)$ define the matrix $S(x)_{i,j} = x_{i+j \mod(n)}$.
$$ S(x) = \left( \begin{array}{rrrr} x_{0} & x_{1} & \cdots & x_{n} \\ x_{1} & x_{2} & \cdots & x_{0} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n} & x_{0} & \cdots & x_{n-1} \end{array} \right). $$
This matrix has a definition which is similiar to http://en.wikipedia.org/wiki/Circulant_matrix but shifts the other way, this is the closest thing I could find which is similiar to the above matrix.
I am using it like so: given $s$ and $a$ where both are column vectors of height $n+1$. Let $p:=(s_0a_0, \ldots s_{n}a_n)$ then let $\hat s, \hat a, \hat p$ denote there Fourier transforms. It follows that
$$\hat p = S(\hat s)\hat a$$
It would be helpful if this matrix had a name so I could Google it. Does anyone have any information on it?
I was hoping to generalise the idea of the above equation to two dimensional Fourier transforms, has anyone got any ideas/links/information to this?