To the theory of discrete boundary value problems

We consider discrete analogues of pseudo-differential operators and related discrete equations and boundary value problems. Existence and uniqueness results for special elliptic discrete boundary value problem and comparison for discrete and continuous solutions are given for certain smooth data in discrete Sobolev– Slobodetskii spaces.


Introduction
As a rule the classical pseudo-differential operator in Euclidean space R m is defined by the formula [1,2]: where the sign $ over a function denotes its Fourier transform, and the functionÃðx; nÞ is called a symbol of a pseudo-differential operator A.
Our main goal here is describing a periodic variant of this definition and studying its certain properties related to solvability of corresponding equations in canonical domains of an Euclidean space. In this paper the main result is related to a comparison of discrete and continuous solutions. We try to preserve maximal correspondence for discrete and continuous cases under digitization, it permits to find more appropriate constructions.
This problem is very large and in our opinion it should include the following aspects according to a lot of physical and technical applications of such operators and related equations: finite and infinite discrete Fourier transform as a natural technique for such equations; choice of appropriate discrete functional spaces; studying solvability for infinite discrete equations; studying solvability of approximating finite discrete equations; a comparison between continuous and infinite discrete equations; a comparison between infinite discrete and finite discrete equations. This is not completed list of questions for studying which we intend to consider. Some results in this direction were obtained for simplest pseudo-differential operators (Calderon-Zygmund operators [3,4]) and corresponding equations. Also certain results are related to approximate solutions.
There are few variants of the theory of discrete boundary value problems (see, for example [5,6]), but these theories are related especially to partial differential operators and do not use the harmonic analysis technique. Since the classical theory of pseudo-differential operators is based on the Fourier transform we will use the discrete Fourier transform and discrete analogue of pseudo-differential operators which will include discrete analogues of partial differential and some integral convolution operators.
2 Discrete spaces and digital operators 2.1 Discrete Sobolev-Slobodetskii spaces Given function u d of a discrete variablex 2 hZ m , h > 0, we define its discrete Fourier transform by the series: where T m ¼ ½Àp; p m ; h ¼ h À1 , and partial sums are taken over cubes, We will remind here some definitions of functional spaces [7] and will consider discrete analogue of the Schwartz space ðe ÀihÁn k À 1Þ 2 and introduce the following.

Definition 1.
The space H s ðhZ m Þ is a closure of the space SðhZ m Þ with respect to the norm, Fourier image of the space H s ðhZ m Þ will be denoted byH s ð hT m Þ.

Digital pseudo-differential operators
One can define some discrete operators for such functions u d . IfÃ d ðnÞ is a periodic function in R m with the basic cube of periods hT m then we consider it as a symbol. We will introduce a digital pseudo-differential operator in the following way.

Definition 2.
A digital pseudo-differential operator A d in a discrete domain D d is called the operator [7], We use the class E a ; a 2 R; [7] with the following condition: and universal positive constants c 1 , c 2 .
Let D & R m be a domain. We will study the equation: in the discrete domain D d D \ hZ m and will seek a solution . Earlier some canonical domains [8][9][10][11][12][13][14][15][16] were considered but in this paper we will discuss the cases R m ; R m þ .
3 Solvability and digital-periodic projectors

Periodic factorization
This case R m þ is very different from R m , and an ellipticity condition is not sufficient for a solvability. A principal role for the solvability takes a concept of the periodic factorization which is defined for an elliptic symbol.
To describe a solvability picture for equation (3) we introduce the following notations. Let us denote We will use a special periodic factorization of an elliptic symbol A d ðnÞ 2 E a : where the factors A d;AE ðnÞ have some analytical properties in half-strips hP AE and satisfy certain estimates [7,8].

Approximation schemes
We will consider the pseudo-differential equation: and suggest for its solution some computational schemes. We assume that the symbol AðnÞ of the operator A satisfies the condition: and it is well-known such a symbol admits factorization, with respect to the last variable n m with the index ae [1]. Since we know solvability conditions for pseudo-differential equations in R m and R m þ [1] we will select such discrete pseudo-differential operators which reserve all needed properties of their continuous analogues.

Equations in a whole space
Let P h be a restriction operator on hZ m , i.e. for u 2 SðR m Þ ðP h uÞðxÞ ¼ uðxÞ; x ¼x 2 hZ m ; 0; x 6 2 hZ m :

&
We tried this projector for simplest pseudo-differential operators, namely Calderon-Zygmund operators, these operators can be treated as pseudo-differential operators of order 0, and we obtained very acceptable results [3,[10][11][12]. But now we will use another restriction operator.
A construction for the restriction operator Q h for functions u 2 SðR m Þ is the following. We take the Fourier transform uðnÞ, then its restriction on hT m and periodically continue it onto a whole R m . Further we apply the inverse discrete Fourier transform F À1 d and obtain a discrete function which is denoted by ðQ h uÞðxÞ;x 2 hZ m . In our opinion the projector Q h is more convenient than P h although the projectors P h and Q h are almost the same according to the following result. where the constant C depends on u only.
Further, the symbol A d ðnÞ will be defined in the following way. We take a restriction of AðnÞ on the cube hT m and periodically extend it onto a whole R m . We consider such h-operator as an approximate operator for A. So, to find a discrete solution for equation (3) for D ¼ R m we can use the following discrete equation: Its solution is given by the formula, so that we do not need to find an approximate solution for an infinite system of linear algebraic equations like [3,10]. For our case we need to apply any kind of cubature formulas for calculating the latter integral and a cubature formula for calculating the Fourier transformṽðnÞ.
According to Lemma 1 one can compare discrete and continuous solutions for enough smooth right-hand sides and symbols. Proofs for Lemma 1 and Theorem 1 are given in [17].

Equations in a half-space
If we put strong enough restrictions on a right-hand side and factorization elements then one can give a comparison between discrete and continuous solutions.

Lemma 2.
If u 2 SðR m Þ then the following estimate: holds for 8b > 0, and the constant C depends on u only. Starting from Lemma 2 and the Theorem 1 we are able to compare discrete and continuous solutions in a half-space. Below we give this comparison under such conditions when a unique solution exists.
To formulate the following theorem we will describe how we need to choose a right-hand side for solving equation (3). We have the following solution of equation (5): where P n 0 ¼ 1 2 ðI þ H n 0 Þ is a projector defined by the classical Hilbert transform with respect to a variable n m [1], 'v is a continuation of v from R m þ into R m in corresponding functional space. Since the right-hand side in equation (5) is defined in hZ m þ then we choose Q h ð'vÞ instead of 'v d to obtain the required estimate. for arbitrary b > 0. One can find proofs for Lemma 2 and Theorem 2 in [17].

Non-trivial case
We have non-uniqueness of a solution for equation (3) for the case À s ¼ n þ d; n 2 N; jdj < 1=2. We consider here the case n = 1.
To obtain the unique solution one needs some additional conditions. Discrete analogues of Dirichlet or Neumann conditions give a very simple case. We will consider here the discrete Dirichlet condition: where g d is a given function of a discrete variable in the discrete hyper-plane hZ mÀ1 . The condition (8) in Fourier images takes the form: and according to the previous theorem we obtain the following integral equation with respect to the unknownc 0 ðn 0 Þ, where we have used the following notation, where X 1 ðnÞ is a polynomial of order 1 of variablesf k ¼ hðe Àihn k À 1Þ, k = 1,. . ., m from the class E 1 . Let us denote,  and after inverse discrete Fourier transform it will be the following, where the function G d ðxÞ of a discrete variable is defined as inverse discrete Fourier transform of the function, b À1 d ðn 0 ÞA À1 d;þ ðn 0 ; n m Þ: The formula (9)

Comparison
To obtain some comparison between discrete and continuous solutions we will remind how the continuous solution looks. The continuous analogue of the discrete boundary value problem is the following: If the index of factorization equals to ae and aeÀs = 1 + d, |d| < 1/2 then the unique solution for the problem (10), (11) is constructed by the similar formula:ũ where, assuming that bðn 0 Þ 6 ¼ 0; 8n 0 2 R mÀ1 . Let us note that this is simplest variant of Shapiro-Lopatinskii condition [1].
We have the following discrete solution:ũ in which we choose special approximations. We take g d ¼ Q h g and A d;AE ðn 0 ; n m Þ we take as restrictions of A AE ðn 0 ; n m Þ on hT m . Then the periodic symbol, satisfies all conditions of periodic factorization with the same index ae. Moreover,g d ðn 0 Þ and A d;þ ðn 0 ; n m Þ coincide with gðn 0 Þ and A þ ðn 0 ; n m Þ respectively on hT m .
Theorem 4. Let ae > 1. IfgðnÞ is a bounded function then a comparison between solutions of problems (3), (8) and (10), (11) is given in the following way, jũðnÞ Àũ d ðnÞj Ch À1 ; n 2 hT m : Proof. For this case we have exact formulas for both continuous and discrete solutions. We will estimate the difference, uðnÞ Àũ d ðnÞ; n 2 hT m : we have, so that we need to estimate b À1 ðn 0 Þ À b À1 d ðn 0 Þ. Since, because infjbðn 0 Þj ! c 3 ; inf jb d ðn 0 Þj ! c 4 then we will estimate the latter difference. Simplest considerations lead to the following estimate, Therefore ifgðn 0 Þ is a bounded function we have the required estimate.

Conclusion
This paper is one of first steps for studying discrete boundary value problems and their connections with classical theory of boundary value problems for elliptic pseudo-differential equations. We intend to study more general situations in forthcoming papers and to obtain approximation estimates for comparison of discrete and continuous solutions.