Some 2-D discrete models of the Bittner operational calculus (Q1848163)

Let \(L^0\) and \(L^1\) be linear spaces such that \(L^1\subset L^0\). Let \(S\in L(L^1,L^0)\), \(T_q\in L(L^0,L^1)\) and \(s_q\in L(L^1,L^1)\), \(q\in Q\), such that \(ST_qu=u\), \(u\in L^0\), and \(T_q Sx=x-s_qx\), \(x\in L^1\). Then the system \((L^0,L^1,S,T_q,s_q,Q)\) is refered as Bittner's operational calculus which generalizes Mikusinski's operational calculus. The author of this paper constructs difference models for the Bittner operational calculus but for doubly indexed sequences \(\{x_{i,j}\}\) taking for the operation \(S:S(x_{i,j})=\{x_{i+1,j+1}\}\) or \(S(x_{i,j})=\{x_{i+1,j+1}-x_{i,j}\}\). Some examples show possible applications of the results to some partial difference equations.