The background of my request is the following: I would need to implement a component that performs an operation f(E) to an arbitrary signal E. The signal is a function of the space E(x), with x going from $-\infty$ to $+\infty$ and the hardware component may occupy all the space. For the sake of simplicity, I discretize the signal into equispaced pixels $x_l$, with $-\infty<l<+\infty$.
Now, this idea turns into the following practical problem: I need to understand if there exists a matrix $A_{k,j}$ so that:
\begin{equation} \sum_{k,j=-\infty}^{+\infty}A_{k,j}E(x_{j+l-k})=f(E(x_{l})). \end{equation} for every $l$ satisfying $-\infty<l<+\infty$.
As a starting simple case, $f$ may be an operator that returns (i) a constant C, or (ii) the heaviside function $f(E(x_{l}))=H(E(x_{l+1})-E(x_{l}))$. Is there somebody that can tell me if that matrix A exists?
Once I know that such $A$ exists, I have to find it and finally, to perform inverse design to engineer the hardware component. But before putting effort in searching for this matrix, I need to know if it exists.
I here underline that the matrix A should perform that operation f on any arbitrary signal E because - just to stay to my example - I want the hardware component to perform an heaviside operation applied to the signal whatever is the signal. Furthermore, if it helps answering my question, I specify that the limits of the space have not to be exactly $\pm \infty$, but they can be also finite, but very big.
Can you help me? Thank you very much for your attention. Best Regards.