A System of Infinite Linear Equations Suppose that $\{a_{i}\}_{i=-\infty}^{\infty}$ with $\sum_{i=-\infty}^\infty a_{i} \lt \infty$ is known and that $\{b_i\}_{i=-\infty}^{\infty}$ is such that
$$\sum_{i=-\infty}^\infty a_{i}b_{-i} =1,$$
and that, for all $k \in \mathbb Z/\{0\}$,
$$\sum_{i=-\infty}^\infty a_{i}b_{-i+k} =0.$$
Is it possible to solve for $\{b_{i}\}_{i=-\infty}^{\infty}$ as a function of $\{a_{i}\}_{i=-\infty}^{\infty}$? 
I just can't figure out where to start from. 

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 $A^{-1}$.
