In numerical integration of a differential equation, a standard approach is to replace it by a suitable difference equation whose solution can be obtained in a stable manner and without troubles from round off errors. However, often the qualitative properties of the solutions of the difference equation are quite different from the solutions of the corresponding differential equations.

For a given differential equation, a difference equation approximation is called *best* if the solution of the difference equation exactly coincides with solutions of the corresponding differential equation evaluated at a discrete sequence of points. Best approximations are not unique (cf. [1, Section 3.6]).

In the recent paper [2] (see also [1]), various discretizations of the harmonic oscillator equation are compared. A best approximation is given by

where denotes the forward difference operator of the first order, that is, for each and On the other hand, in the article [3], a characterization of -maximal regularity for a discrete second-order equation in Banach spaces was studied, but without taking into account the best approximation character of the equation. From an applied perspective, the techniques used in [3] are interesting when applied to concrete difference equations, but additional difficulties appear, because among other things, we need to get explicit formulas for the solution of the equation to be studied.

We study in this paper the discrete second-order equation

on complex Banach spaces, where . Of course, in the finite-dimensional setting, (1.2) includes systems of linear difference equations, but the most interesting application concerns with partial difference equations. In fact, the homogeneous equation associated to (1.2) corresponds to the best discretization of the wave equation (cf. [1, Section 3.14]).

We prove that well posedness, that is, maximal regularity of (1.2) in vector-valued spaces, is characterized on Banach spaces having the unconditional martingale difference property ( see, e.g., [4]) by the -boundedness of the set

The general framework for the proof of our statement uses a new approach based on operator-valued Fourier multipliers. In the continuous time setting, the relation between operator-valued Fourier multiplier and boundedness of their symbols is well documented (see, e.g., [5–10]), but we emphasize that the discrete counterpart is too incipient and limited essentially a very few articles (see, e.g., [11, 12]). We believe that the development of this topic could have a strong applied potential. This would lead to very interesting problems related to difference equations arising in numerical analysis, for instance. From this perspective the results obtained in this work are, to the best of our knowledge, new.

We recall that in the continuous case, it is well known that the study of maximal regularity is very useful for treating semilinear and quasilinear problems. (see, e.g., Amann [13], Denk et al. [8], Clément et al. [14], the survey by Arendt [7] and the bibliography therein). However it should be noted that for nonlinear discrete time evolution equations some additional difficulties appear. In fact, we observe that this approach cannot be done by a direct translation of the proofs from the continuous time setting to the discrete time setting. Indeed, the former only allows to construct a solution on a (possibly very short) time interval, the global solution being then obtained by extension results. This technique will obviously fail in the discrete time setting, where no such thing as an arbitrary short time interval exists. In the recent work [15], the authors have found a way around the "short time interval" problem to treat semilinear problems for certain evolution equations of second order. One more case merits mentioning here is Volterra difference equations which describe processes whose current state is determined by their entire prehistory (see, e.g., [16, 17], and the references given there). These processes are encountered, for example, in mathematical models in population dynamics as well as in models of propagation of perturbation in matter with memory. In this direction one of the authors in [18] considered maximal regularity for Volterra difference equations with infinite delay.

The paper is organized as follows. The second section provides the definitions and preliminary results to be used in the theorems stated and proved in this work. In particular to facilitate a comprehensive understanding to the reader we have supplied several basic -boundedness properties. In the third section, we will give a geometrical link for the best discretization of the harmonic oscillator equation. In the fourth section, we treat the existence and uniqueness problem for (1.2). In the fifth section, we obtain a characterization about maximal regularity for (1.2).