The maximum principle and growth estimates for partial difference equations

Summary.

Linear equations in partial forward and backward differences for functions in n independent variables (partial difference equations or PΔE’s) on arbitrary non-finite subsets $A \subset \mathbb{Z}^{n}$ are considered. To estimate the growth of the solution, its recursive construction is used. The existence and uniqueness theorem for initial value problems guarantees, on one hand, that such a recursive construction is always possible and, on the other hand, such a construction allows to prove a statement, which we call the maximum principle: under certain conditions the values of the solution are bounded by the norm of the initial values (in the homogeneous case), or bounded by the norm of the input (in the non-homogeneous case with zero initial values). These conditions, demanding that the absolute value of a certain coefficient is dominant, i.e greater or equal to the sum of absolute values of the remaining coefficients, are easy to verify. Such a maximum principle is further applied to obtain various growth estimates. Examples include growth estimates for Stirling numbers, Euler numbers, coefficients of orthogonal polynomials and other sequences defined by PΔE’s. Generalizations for systems of PΔE’s and some nonlinear PΔE’s are also formulated.