8. On the Cauchy Problem for Multidimensional Difference Equations in Rational Cone

Abstract

The Cauchy problem for multidimensional difference equations in rational cone is formulated and sufficient condition for its solvability is given. The notion of multisection of multiple Laurent series with the support in a rational cone is defined. The formulae which express any multisection through original series are presented.

On Formal Solutions of H¨ormander’s Initial-boundary Value Problem in the Class of Laurent Series

“…To formulate an analog of the initial-boundary value problem of Hörmander, we need the definitions of a rational cone, of the ring of Laurent series with supports in these cones and derivations of the ring of such series (see [6,7]).…”

“…c ω ζ ω is called the characteristic polynomial for the polynomial differential operator (6). By the order d ν of P (D), we mean the weighted homogeneous degree deg ν P (ζ) of the characteristic polynomial, i.e., d ν = max ω∈Ω |ω| ν .…”

“…Let m ∈ Ω and let |m| ν = d be the order of a differential operator. If the coefficients of operator (6) are constant and the coefficients of the principal symbol of P d (D) satisfy the condition…”

Evaluating the rational generating function for the solution of the Cauchy problem for a two-dimensional difference equation with constant coefficients

“…For n > 1 the proof was given in [12]. The properties of generating function of solutions of a difference equation in rational cones of integer lattice were studied by T. Nekrasova (see, e.g., [17]).…”

Constant coefficient linear difference equations on the rational cones of the integer lattice

“…Remark 1. Formula (16) is proved in [10] in the case K ∩ Z n = Z n + , while in [11] for the intersection of K with the sublattice Λ of the lattice Z n generated by some vectors a 1 , . .…”