I would like to solve in $f$ the equation $f(x) - f(x-d) = g(x),$ where $g$ is a given function and $d$ a given constant delay. We can assume $f(x) = 0$ for negative $x$.
When $g$ is sampled (with a unit step), are there known numerical approaches to estimate the corresponding $f$ samples ? $d$ is not necessarily integer, but in practice is a small value (say 5 to 25).
I think this can be called an "(ordinary) finite-difference equation", but I am having a hard time to find relevant documentation, as the topic is swamped by theory on recurrence relations and solutions of differential equations by discretization.
Since $f(x) = 0$ for negative $x\,$ you'll get :
and more generally :
$$f(x)=\sum_{k=0}^n g(x-k\,d)\quad\text{for $\,x\,$ in}\;\;(n\,d,\,(n+1)\,d)$$
In your specific case ($g$ sampled with a unit step) this will only work for $d$ integer.
I would suggest linear interpolation between the two nearest integer values if $\,x-k\,d\,$ is not integer (of course higher degree interpolations are possible but seem more 'risky'...).