Residue current approach to Ehrenpreis–Malgrange type theorem for linear differential equations with constant coefficients and commensurate time lags

Abstract

We introduce a ring H of partial difference-differential operators with constant coefficients initially defined by H. Glüsing-Lürßen for ordinary difference-differential operators and investigate its cohomological properties. Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last section, we deduce from this injectivity properties of various function modules over H as well as the density results of exponential polynomial solutions for partial difference-differential equations.

The flatness of the 𝒪-module of smooth functions and integral representation

“…The idea to use integral formulas to find holomorphic solutions to this kind of equations, often referred to as division problems, was introduced by Berndtsson in [11]. It was further developed and adapted for a variety of situtions, see, e.g., [2,4,5,6,7,8,9,10,12,15,16,17,18].…”

The flatness of the $\Ok$-module of smooth functions and integral representation

“…The idea to use integral formulas to find holomorphic solutions to this kind of equations, often referred to as division problems, was introduced by Berndtsson in [10]. It was further developed and adapted for a variety of situtions, see, e.g., [20,12,13,14,18,19,21,1,4,5,7,8,9].…”