A System of Infinite Linear Equations

Note: This did not help, but was fun:

Let us try this for a finite sum: $$ 1 = \begin{pmatrix} a_{-1} & a_0 & a_1 \end{pmatrix} \begin{pmatrix} b_1 \\ b_0 \\ b_{-1} \end{pmatrix} = \begin{pmatrix} a_{-1} & a_0 & a_1 \end{pmatrix} \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} b_{-1} \\ b_0 \\ b_1 \end{pmatrix} $$ For $k=1$: $$ 0 = \begin{pmatrix} a_{-1} & a_0 & a_1 \end{pmatrix} \begin{pmatrix} 0 \\ b_1 \\ b_0 \end{pmatrix} = \begin{pmatrix} a_{-1} & a_0 & a_1 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} b_{-1} \\ b_0 \\ b_1 \end{pmatrix} $$ This leads to the equations $$ a^\top D^{(k)} b = \delta_{k0} $$ for a matrix $D^{(k)}$ with components $$ d_{ij}^{(k)} = \delta_{(k-i)j} $$ where we also use negative indices for the matrix elements, thus having the center element at $d_{00}^{(k)}$.

This leads to a matrix equation $$ A b = y $$ with $$ A = \begin{pmatrix} & \vdots & \vdots & \vdots \\ \dotsb & a_0 & a_{-1} & a_{-2} & \dotsb \\ \dotsb & a_1 & a_0 & a_{-1} & \dotsb \\ \dotsb & a_2 & a_1 & a_{0} & \dotsb \\ & \vdots & \vdots & \vdots \\ \end{pmatrix} \quad y= \begin{pmatrix} \vdots \\ 0 \\ 1 \\ 0 \\ \vdots \end{pmatrix} $$ and the formal solution $$ b = A^{-1} y $$ which is the central column of $A^{-1}$.