Two-Scale Estimates for Special Finite Discrete Operators

We consider a certain finite discrete approximation for multidimensional Calderon–Zygmund integral operator and give a comparison between solutions of corresponding equations in some spaces of discrete functions.


Introduction
We consider the following Calderon-Zygmund singular integral operator [10] (Ku)(x) ≡ v.p. R m K(x − y)u(y)dy, x ∈ R m , (1.1) in the Lebesgue space L 2 (R m ) and its discrete analogue of the following type in the space L 2 (hZ m ) ≡ l 2 of functions u d of a discrete variablex ∈ hZ m . We recall [10] that that a symbol σ(ξ) of the operator K is the Fourier transform of its kernel in principal value sense σ(ξ) = lim and such symbol is called an elliptic symbol if inf ξ |σ(ξ)| > 0.
Our main goal is the following. Starting from the operator (1.1) we introduce the discrete operator (1.2) acting in infinite dimensional space so that it preserves basic properties of the operator (1.1) related to an ellipticity and invertibility [18,19,20,21,22]. Further to obtain computational algorithms we would like to construct finite dimensional analogue of the operator (1.2) preserving same properties, and to obtain results on comparison of these operators and solutions of corresponding equations.

Some previous approaches and studies
In books [2,3,5,6,14,15,17] authors present studies on convolution equations, one-dimensional singular integral equations and multidimensional weakly singular integral equations related to approximate solution of these equations.
As usual there are the following questions in these studies: 1) to find an approximate equation desirable in a finite-dimensional (N -dimensional) space so that under enough large N this approximate equation will be uniquely solvable in an appropriate space; 2) to obtain an error estimate between exact solution of an initial equation and exact solution of approximate equation in dependence on N . As far as we know these questions were investigated fully for one-dimensional singular integral equations on smooth curves L in a complex plane C [8,15] convolution equations with integrable kernel K(x) on a straight line and some their multidimensional analogues [2,3,5] and for multidimensional integral equations with a weak singularity where D is a domain with a smooth boundary ∂D and the kernel K(x, y) satisfies the estimate last equations are generated by compact integral operators in appropriate spaces [17].
An algebraic approach based on a local principle for studying certain finite approximations for integral operators was used in many papers, and its development for last years is presented in books for example [2,3,6]. But this method of C * -algebras permits to prove a solvability of approximating equation but it can't help for obtaining an error estimate for solutions of these equations. Moreover concrete applications of the method are related as a rule to one-dimensional singular integral operators and equations.
For an error estimate there are some computational results [9,12,13,16], and a lot of results are related to special equations for applied problems (see for example [7]). Multidimensional case is more complicated because there is no such advanced theory similar classical Riemann boundary value problem [4,11].
We would like to note that in general there are no valuable results for approximate solution of multidimensional singular integral equations with Calderon -Zygmund operators although such equations in distinct form arise in many problems of partial differential equations and mathematical physics [10]. In our opinion a direct digitization is more convenient for computer calculations than other methods, and we try to use and justify this approach.

Definitions and notations
We accept the following conventions and notations. The kernel K(x) is a differentiable function on the unit sphere in R m , K(0) ≡ 0, and K d is a restriction of the kernel K on lattice points hZ m . Let h be a size of a "spatial quant", N be a size of our "Universe". These parameters will be tend to zero and infinity respectively. For the operator (1.1) we'll use the following reduction. First we replace the operator (1.1) by the discrete operator (1.2), and second we approximate this series by a special finite sum (2.1) For K d,N in the formula (2.1) we suggest a following construction. If K d is a restriction of the continual kernel K on lattice points hZ m then we take a restriction of the K d on points hZ m ∩ Q N , where and denote by K d,N its periodic continuation on the whole hZ m .
Let's denote by P N the restriction operator hZ m −→ hZ m ∩ Q N , and the space L 2 (hZ m ∩ Q N ) is denoted by l 2 N so that P N is a projector l 2 → l 2 N . Definition 1. Approximation rate of operators K d and K d,N is called the following operator norm It is non-trivial to obtain an estimate for the operator norm, but we'll give an estimate for an individual element assuming the existence of some its properties. More precisely we'll suppose that the element u d is a restriction of the function u which has Hölder property in R m and u(x) = o(|x| γ ), |x| → ∞, with some γ > 0.
We will obtain a "weak estimate" for approximation rate but enough for our purposes. We assume additionally that a function u d is a restriction on hZ m of continuous function with certain estimates [20,21]. Let's define the discrete space C h (α, β) as a functional space of discrete variablex ∈ hZ m with finite norm It means that the function u d ∈ C h (α, β) satisfies the following estimates Let us note that under required assumptions C h (α, β) ⊂ L 2 (hZ m ), and these discrete space are discrete analogue of corresponding subspaces of continuous functions [1].
Proof. Let us write where I is an identity operator in L 2 (hZ m ).
First two summands have annihilated, and we need to estimate only the last summand. We have because norms of operators K d are uniformly bounded, and for the last norm taking into account above estimates we can write The last integral using spherical coordinates gives the estimate N m+2(α−β) which tends to 0 under n → ∞ if β > α + m/2. Remark 1. Similar theorem was obtained in [19,20] for the space C h (α, β).
This theorem plays a key role for obtaining an estimate for approximate solution of an multidimensional singular integral equation with the operator (1.1) and permits to use fast Fourier transform for evaluating a numerical solution. Some test calculations were given in [20].

Discrete Fourier transform
We define the discrete Fourier transform for a function u d of a discrete variablẽ x ∈ hZ m as the series This discrete Fourier transform has same properties like standard continual Fourier transform, particularly for a discrete convolution of two discrete we have the well known multiplication property If we apply this property to the operator K d we obtain Let us denote (F d K d )(ξ) ≡ σ d (ξ) and give the following Definition 2. The function σ d (ξ), ξ ∈ T m , is called a periodic symbol of the operator K d .
We will assume below that the symbol σ d (ξ) ∈ C( T m ) therefore we have immediately the following So we see that an arbitrary elliptic periodic symbol σ d (ξ) corresponds to an invertible operator K d in the space L 2 (hZ m ).

Finite discrete Fourier transform
Definition 4. For a function u d,N ∈ l 2 N its finite discrete Fourier transform is defined by the formulã Definition 5. A symbol of the operator K d,N is called the function σ d,N (ξ) of a discrete variableξ ∈ Z m ∩ Q N defined by the formula

Comparison between infinite and finite discrete operators
Theorem 2. If the operator K d is invertible in the space l 2 then the operator K d,N is invertible in the space l 2 N for enough large N .
Proof. Let the function is a segment of the Fourier series For the proof see [19]. Proof. Using the Theorem 2 and the property that the norm of the operator K d,N is equivalent to max ξ∈ Z m ∩Q N |σ d,N (ξ)| (see also [10]) we obtain the required assertion.

Correlation between N, h and location ofx
The proved Theorem 1 is a very rough estimate. It is possible to obtain more exact error estimate for finite discrete solution taking into account a location of the pointx and relations between parameters N and h. We consider three types of equations where P d is a restriction operator which given continuous function v defined on R m maps to a collection of its values on Z m , and and would like to have an estimate for nearness of their solutions. Let us denote by r(x) the distance betweenx ∈ hZ m ∩ Q N .
in other cases holds, c 1 , c 2 are constants non-depending on h, N .
Proof. Assuming N is enough large so that both operators K d and K d,N are invertible in spaces l 2 and l 2 N respectively let us consider the difference where We'll consider summands separately and give pointwise estimates for them. For we have I 2 (x) = 0 because this difference is not zero for points from hZ m \ Q N only, so we need to estimate I 1 (x) only. First we need to say some words on a structure of operators K −1 d and K −1 d,N which we have constructed for operators K d : l 2 → l 2 and K d,N : l 2 N → l 2 N . Lemma 1 [19] implies that the norm of the operator K d doesn't depend on h. Also we have an analogue assertion for the operator K d,N (Lemma 2). Moreover the operator K −1 d is generated by Calderon-Zygmund operator with symbol σ −1 (ξ) and corresponding kernel K −1 (x), so that the kernel K −1 d (x) of the discrete operator K −1 d is a restriction of the kernel K −1 (x) on the lattice hZ m .
Further the operator K −1 d,N is constructed by the same way. We take the discrete kernel K −1 d (x) then we take its restriction on Q N and a periodic continuation on a whole hZ m . A symbol of a such operator will be σ −1 d,N (ξ), and we conserve all required properties.
Let us start to estimate I 1 (x).
and we need to estimate the last sum only.
We'll represent I 1 (x) as a sum I 1 (x) = I 11 (x) + I 12 (x), where Then and for enough small h we have where D N (x) is a ball with a center inx and radius ∼ N . Using spherical coordinates with a center inx we obtain For I 12 (x) we use another estimate. We write For I 121 (x) we have |x −ỹ| ∼ |x|, and consequently |I 121 (x)| ≤ cN α−β . For I 122 (x) we have |x −ỹ| ∼ |ỹ|, and then 2) r(x) ∼ N . Here we have |x −ỹ| ∼ N . Thus Since we are interested in small h the last sum can be dominated by the following integral R m \Q N |y| α−β dy and calculations with spherical coordinates give the estimate for β − α > 0 3) r(x) ∼ 1. So we have the following estimate because |x −ỹ| ∼ 1 + |ỹ|. Since we are interested in small h the last sum can be dominated by the following integral

Conclusions
Our considerations give a certain algorithm for solving a simplest singular integral equation in a whole space R m , and also in R m + taking into account authors' conclusions in [21,22]. The error estimate for finite discrete solutions shows that varying N, h we can obtain a necessary sharpness. Collecting all authors' results [18,19,20,21,22] related to equations (4.1), (4.2), (4.3) we conclude that there is a certain correspondence between solvability of these equations and their solutions.