Maximal Regularity of the Discrete Harmonic Oscillator Equation
© Airton Castro et al. 2009
Received: 31 October 2008
Accepted: 9 February 2009
Published: 8 March 2009
We give a representation of the solution for the best approximation of the harmonic oscillator equation formulated in a general Banach space setting, and a characterization of -maximal regularity—or well posedness—solely in terms of -boundedness properties of the resolvent operator involved in the equation.
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]).
where denotes the forward difference operator of the first order, that is, for each and On the other hand, in the article , 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  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.
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]).
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 , Denk et al. , Clément et al. , the survey by Arendt  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 , 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  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).
Let and be the Banach spaces, let be the space of bounded linear operators from into . Let denote the set of nonnegative integer numbers, the forward difference operator of the first order, that is, for each and We introduce the means
for all The least such that (2.2) is satisfied is called the -bound of and is denoted
An equivalent definition using the Rademacher functions can be found in . We note that -boundedness clearly implies uniformly boundedness. In fact, we have that If , the notion of -boundedness is strictly stronger than boundedness unless the underlying space is isomorphic to a Hilbert space [5, Proposition 1.17]. Some useful criteria for -boundedness are provided in [5, 8, 19]. We remark that the concept of -boundedness plays a fundamental role in recent works by Clément and Da Prato , Clément et al. , Weis [9, 10], Arendt and Bu [5, 6], as well as Keyantuo and Lizama [22–25].
Let be -bounded sets, then is - bounded.
- (b)Let and be -bounded sets, then is - bounded and(2.3)
Also, each subset of the form is - bounded whenever is bounded.
is bounded. For recent and related results on analytic operators we refer to .
3. Spectral Properties and Open Problems
which reminds the "symmetric" version of Euler's discretization scheme, but that appears in the discretization of the second derivative is replaced by .
If in (3.1), then we have a well-defined recurrence relation of order in case or (and of order ) in case .
In case , we have , that is, a recurrence relation of order , which can be not well defined unless . Finally, in case , is of order (note that here we need ).
Then, for each fixed, describes a curve in the complex plane such that
The curve attains the minimum length at
From which the conclusion follows.
and that this value exactly corresponds to the step size in the best discretization of the harmonic oscillator obtained in . We conjecture that there is a general link between the geometrical properties of curves related to classes of difference equations and the property of best approximation. This is possibly a very difficult task, which we do not touch in this paper.
In what follows we denote ; and The following result relates the values of with the spectrum of the operator . It will be essential in the proof of our characterization of well posedness for (1.2) in -vector-valued spaces given in Section 5 (cf. Theorem 5.2).
Finally, we observe that
4. Existence and Uniqueness
then . It follows from induction. In fact, suppose that for all , choosing in (4.2) we get .
Also we note that the convolution theorem for the discrete Fourier transform holds, that is, Further properties can be found in [28, Section 5.1]. Our main result in this section, on existence and uniqueness of solution for (4.1), read as follows.
and the proof is finished.
5. Maximal Regularity
Let . One says that (4.1) has discrete maximal regularity if defines a bounded operator .
A similar analysis as above can be carried out when we consider more general initial conditions, but the price to pay for this is that the proof would certainly require additional -summability condition on The following is the main result of this paper.
Let be a UMD space and let analytic. Then the following assertions are equivalent.
Equation (1.2) has discrete maximal regularity.
Explicitly, is given by . We conclude, from [11, Proposition 1.4], that the set in (ii) is -bounded.
Then, by uniqueness of the Fourier transform, we conclude that
Let be a Hilbert space and let be an analytic operator. Then the following assertions are equivalent.
Equation (1.2) has discrete maximal regularity.
We emphasize that from a more theoretical perspective, our results also are true when we consider the more general equation (3.1) instead of (1.1), but additional hypothesis will be needed (cf. Remark 3.1). Until now literature about this subject is too incipient and should be developed.
The authors are very grateful to the referee for pointing out omissions and providing nice comments and suggestions. This work was done while the third author was visiting the Departamento de Matemática, Universidade Federal de Pernambuco, Recife, Brazil. The second author is partially supported by CNPQ/Brazil. The third author is partially financed by Laboratorio de Análisis Estocástico, Proyecto Anillo ACT-13, and CNPq/Brazil under Grant 300702/2007-08.
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