A Laplace-like method for solving linear difference equations (Q797067)

For a given positive number \(\tau\) let \(I_ n\) be the right-open interval \([n\tau,(n+1)\tau),\) \(n\in {\mathbb{Z}}\). Then for \(t\in {\mathbb{R}}\) let \(t_ k\) be the element of \(I_ k\) such that \(| t_ k-t|\) is an integer multiple of \(\tau\). On the set F of functions f:\({\mathbb{R}}\to {\mathbb{C}}\) which vanish to the left of some point (not the same for all mappings) an operation * (discrete convolution) is defined by \[ (f*g)(t)=\sum^{\infty}_{k=-\infty}f(t_{n-k})g(t_ k),\quad t\in I_ n. \] It is shown that with \(+\) in the obvious meaning \(<F,+,*>\) is a commutative ring with complex operators. One of the most important properties of this ring is that it possesses elements a,b such that for a given function \(f\in F\) (which vanishes for every \(t<0) f(t+\tau)- f(t)=a*f-b*f_ 0\) for every \(t\geq 0\), where \(f_ 0(t)=f(t)\) if \(t\in I_ 0\) and 0 elsewhere. From this the applicability to linear difference equations with constant coefficients can be realized. Thus the difference equation is reducable to an algebraic equation. Two examples are given and the connection with the operational methods of Mikusinski and Fenyö is pointed out.