The order of the best transfinite interpolation of functions with bounded laplacian with the help of harmonic splines on box partitions

Показано, що похибка найкращої трансфiнiтної iнтерполяцiї функцiй з обмеженим лапласiаном за допомогою кусково-гармонiчних сплайнiв на прямокутних розбиттях, якi складаються з N елементiв, має порядок N−2, коли N →∞. Ключовi слова: трансфiнiтна iнтерполяцiя, найкраще наближення, кусково-гармонiчний сплайн. Показано, что погрешность наилучшей трансфинитой интерполяции функций с ограниченным лапласианом с помощью кусочно-гармонических сплайнов на прямоугольных разбиениях, состоящих из N элементов, имеет порядок N−2, когда N →∞. Ключевые слова: трансфинитная интерполяция, наилучшее приближение, кусочногармонический сплайн. We show that the error of the best transfinite interpolation of functions with bounded laplacian with the help of harmonic splines on box partitions comprising N elements has the order N−2 as N →∞.

Depending on the information available on the multivariate function a variety of interpolation schemes exist in Approximation Theory.Transfinite interpolation is the special type of interpolation schemes that consists in constructing the function coinciding with the given function along some manifolds.It was first considered by D. Mangeron [1] in 1932 and was developed further in papers [2,3,4,5,6,7,8,9,10,11,12,13] (see also surveys [14,15,16] and references therein).Transfinite interpolation finds applications in computer tomography, finite elements methods, geometric modeling, cartography and many others (see [17,18,19,12]).
In certain important situations we may assume that the manifolds, along which the values of the interpolated function are known, split the function domain of definition into sub-domains with the Lipshitz boundary.The natural method of transfinite interpolation in such case is delivered by harmonic splines that coincide with the harmonic continuation of the function from the boundary of every element in the partition into its interior.Properties of transfinite interpolation by harmonic splines

THE ORDER OF THE ERROR OF THE BEST TRANSFINITE INTERPOLATION
were studied in [8,10,11,12,13], where the problem of the best transfinite interpolation of multivariate functions on box partitions was considered.In this paper we continue the investigation of this problem and find the exact order of error of the best transfinite interpolation with the help of harmonic splines on box partitions.Let us proceed to the rigorous settings of the problem.
Let d ∈ N and Ω ⊂ R d be a compact set.By C(Ω) we denote the standard space of continuous on Ω functions f : Ω → R. For 1 ≤ s ≤ ∞, we denote by L s (Ω) the space of measurable functions f : Ω → R endowed with the standard norm As usually, by ∆ we denote d-variate Laplace operator: For a function f ∈ C(Π) and a box partition P of Π, we construct the harmonic spline S P f in the following way.For every Ω ∈ P , by u = S P f | Ω we denote the solution of the Dirichlet problem for the Laplace equation: From the construction it is easy to see that the spline S P f is continuous on Π and coincide with the function f on the set Ω∈P ∂Ω.For 1 ≤ q ≤ ∞, we consider the class Next, for a box partition P of Π, we define the error of transfinite interpolation of functions f ∈ W ∆ q by harmonic splines: The quantity (1) was studied in papers [10,11] on the special class of box partitions P that consist of d-boxes obtained by splitting Π with the help of a finite number of (d − 1)-variate hyperplanes are orthogonal to the coordinate axes.Repeating the proof of Theorems 1 and 2 in [11], we can establish the following representation for in terms of the norms of the Green function.
Proposition 1.Let P be an arbitrary box partition of Π, 1 ≤ s < ∞, s = s/(s−1).Then , where G Ω is the Green function in the Dirichlet problem for the Laplace equation on the set Ω.Moreover, For N ∈ N, by P N := P N (Π) we denote the set of all box partitions of Π comprising N elements.The problem of the best transfinite interpolation of functions from the class W ∆ q by harmonic splines on box partitions P ∈ P N can be stated as follows.Problem 1. Find the quantity and find optimal partitions P * ∈ P N (if any exists) such that E N W ∆ q s = E P * W ∆ q s .V. F. Babenko and T. Yu.Leskevych [11]  hyperplanes orthogonal to the axis Ox 2 , etc. From definition it is clear that P m ⊂ P |m| .Also, we denote by P * m the regular box partition from P m comprising of |m| equal dboxes.The following proposition was established in [11] (see arguments right above Corollaries 2 and 4 in [11]).Proposition 2. Let m = (m 1 , . . ., m d ) be positive integer multi-index.Then It was natural to expect that the order of the quantity E N W ∆ ∞ s , 1 ≤ s ≤ ∞, degrades with the increase of the number of dimensions, as always happen in the case of approximating isotropic classes of functions with the help of polynomial splines (see [20]).However, in [13] it was established (see Theorem 1) that E N W ∆ ∞ s = O (N −2 ) as N → ∞.Also, it was shown that this order is achieved on box partitions P * ,j N , j = 1, . . ., d, obtained by splitting Π into equal d-boxes with the help of N − 1 hyperplanes orthogonal to the axis Ox j .In this paper we will prove that N −2 is the correct order of quantity E N W ∆ ∞ s as N → ∞.The main result of this paper is contained in the following statement.

THE ORDER OF THE ERROR OF THE BEST TRANSFINITE INTERPOLATION
Theorem 1.Let d ∈ N, 1 ≤ s ≤ ∞ and s = s/(s − 1).Then there exist constants C 1 , C 2 > 0 such that, for every N ∈ N, Proof.By Theorem 1 in [13] there exists a constant C * 2 > 0 such that lim sup Hence, there exists a constant C 2 > 0 such that the right hand inequality in (3) holds true for every N ∈ N.
Let us now establish the lower estimate for N 2 • E N W ∆ ∞ s .First, we consider the case s = ∞.Let P ∈ P N be a box partition of Π.Let us show that some box Ω in P contains a box obtained from 1 N • Π with the help of a translation.To this end we assume to the contrary that, for every Ω ∈ P , there exists index j Ω ∈ {1, . . ., d} such that the length of the side of Ω that is parallel to the axis Ox j Ω does not exceed Then we can estimate the volume of any box Ω ∈ P as follows: However, by definition of the partition P ∈ P N on Π we have which leads to a contradiction.Therefore, some box Ω * ∈ P contains a box Π * obtained from 1 N • Π with the help of a translation.Now, without loss of generality we assume that b 1 = min j=1,...,d b j .Also, let s be the center of the box Π * and B be the ball with the center at the point s and radius b 1 N .In addition, we define a function τ : [0, +∞) → R: Let us consider the function ϕ P : Π → R where: Obviously, ϕ P ∈ W ∆ ∞ and ϕ P vanishes on the boundary of every box Ω ∈ P .Hence, S P ϕ P ≡ 0 and, as a result, we obtain Consequently, Let us consider the case when s ∈ [1, ∞).Let P ∈ P N be a box partition of Π.For every box Ω ∈ P , by a Ω we denote the smallest side of Ω, and by Ω * the set of points in Ω having the distance at least 1 4 a Ω to ∂Ω, i.e.
Let us estimate from below the Lebesgue measure of the set Ω * .To this end we observe that Ω * is the box whose sides are parallel to the sides of Ω.If b is the length of a side of box Ω, then the length of the parallel side of Now, we let P ∈ P N be a partition of Π.Without loss of generality we assume that b 1 = min Also, it is clear that a Ω ≥ b 1 2N , for every Ω ∈ I. Now, for a > 0, we consider the function τ a : [0, +∞) → R:

d.
Let b 1 , . . ., b d > 0 be given and denote by Π = [0, b 1 ] × . . .× [0, b d ] the d-box in R d those sides are parallel to the coordinate axes.A finite set of d-boxes P is called box partition of P if Π = Ω∈P Ω and Ω ∩ Ω = Ø, for every two distinct d-boxes Ω , Ω ∈ P .