A Deﬁnition of Two-Dimensional Schoenberg Type Operators

: In this paper, a way to build two-dimensional Schoenberg type operators with arbitrary knots or with equidistant knots, respectively, is presented. The order of approximation reached by these operators was studied by obtaining a Voronovskaja type asymptotic theorem and using estimates in terms of second-order moduli of continuity.


Introduction
The Schoenberg operators provide a concrete method for obtaining spline approximations of functions. These operators have very good approximation properties. However, they are not very present in the literature. The study of the positive linear Schoenberg operators was the subject of several recent papers, among which we mention here Beutel, Gonska, Kacso and Tachev [1], Tachev [2,3], and Tachev and Zapryanova [4]. In [1] variation-diminishing one-dimensional Schoenberg spline operators, especially with equidistant knots and inequalities in terms of moduli of continuity, were studied. An analysis of the second moment of one-dimensional Schoenberg spline operators moment was presented. A discussion about the degree of simultaneous approximation for multivariate case, more specifically for Boolean sums, was realized. Additionally, a similar discussion was presented for tensor products of one-dimensional Schoenberg spline operators. In [2] a lower bound for the second moment of one-dimensional Schoenberg spline operators is made. In [3] a Voronovskaja's type theorem for one-dimensional Schoenberg spline operators is presented. In [4] a generalized inverse theorem for one-dimensional Schoenberg spline operators is established.
In practice, the use of the one-dimensional Schoenberg operators offers many advantages. This fact was illustrated, for example, in the recent paper [5], in which these operators were applied for improving the clear sky models used to estimate the direct solar irradiance, with influences of the system design and financial benefits.
The subject of multivariate splines was approached by different methods and from various points of view, such as in the papers written by: Curry and Schoenberg [6], Goodman and Lee [7], de Boor and Hollig [8], Karlin et al. [9], Goodman [10], Chui [11], Schumaker [12], Conti and Morandi [13], Ugarte et al. [14], and Groselj and Knez [15]. Curry and Schoenberg indicated in [6] that the multivariate spline functions can be constructed from volumes of slices of polyhedra; therefore, papers can be found that were written toward that direction. For example, this idea led to the recurrence relations for multivariate splines presented by Karlin et al. in [9]. Goodman and Lee in [7] and Goodman in [10] approached the subject of multidimensional Bernstein-Schoenberg operators depending on m-dimensional volume. In [8] the subject of multidimensional B-splines is treated by de Boor and Hollig as the m-shadow of the polyhedral convex body included in R n . In [11] the Box splines, multivariate truncated powers and many other aspects of multivariate splines are studied by Chui. Conti and Morandi used mixed splines to solve the interproximation problem for surfaces in the case of scattered data in [13]. The aim of Ugarte et al. in the paper [14] was to propose different possibilities of modeling the space-time interaction using one dimensional, two-dimensional and three dimensional B-splines. In [15] Groselj and Knez introduced the notion of a balanced 10-split for the construction of non-negative basis functions for the space of C 1 quadratic splines and showed that the considered split has potential to be used for the construction of C 2 splines.
The aim of our paper is to consider a new approach in spline approximation in the two-dimensional case, based on a two-dimensional version of Schoenberg operators. The two-dimensional Schoenberg type operators are constructed by generalization of the one-dimensional Schoenberg operators. As a result of this generalization we reach a particular form of tensor-product B-splines. The subject of the tensor-product B-splines is treated in several papers, for example in [12].

Two-Dimensional Schoenberg Type Operators on Arbitrary Nodes
We define two-dimensional Schoenberg type operators as follows. Let us consider the knot sequence ∆ n,h where n > 0, h > 0 and the knot sequence ∆ m,k where m > 0, k > 0. The Greville abscissas associated with knot sequence ∆ n,h are and the Greville abscissas associated with knot sequence ∆ m,k are The B-splines N i,h (α) depending on ∆ n,h are defined in the following mode: and the B-splines N j,k (β) depending on ∆ m,k , by: and N j,k (β) = 0, for j < r − k or j ≥ r + 1, and N j,k (α) ≥ 0, for r − k ≤ j ≤ r.
The following relations are well known: Analogous relations are fulfilled for N j,k . This notation- Definition 1. Two-dimensional Schoenberg type operator associated with∆ has the form where ϕ :
Proof. We assume that condition (10) Let n ε , m ε ∈ N such that: Then, for such n and m, we have The norm of the knot sequence∆ is given by where A quantitative version of the degree of approximation can be given using the first order modulus of continuity, defined as follows: where ϕ ∈ C([0, 1] 2 ), ρ > 0.
In [16], the subject of the second moment of variation-diminishing splines is approached. The second moment of the second degree Schoenberg one-dimensional operators was established in [17] and of the third degree Schoenberg one-dimensional operators in [18]. Further on, the form of second moment of two-dimensional Schoenberg type operators with h = k = 3 is presented.

Theorem 3. The second moment of the two-dimensional Schoenberg type operators S∆ for h
Proof. By applying the linearity, Proposition 2(iii) and (iv) follow and the result given in [18].

Two-Dimensional Schoenberg Type Operators with Equidistant Knots
Now the case h = k = 3, m = n and equidistant knots is analyzed. More precisely, the equidistant knots are α i = i n , 0 ≤ i ≤ n, and the extra-knots are The Greville abscissas are in this case The B-splines are respectively Two-dimensional Schoenberg type operators, with h = k = 3 and m = n, with equidistant knots is denoted byS n,3 : and the one-dimensional k degree Schoenberg operators with equidistant knots are denoted by S n,k .
Moduli of continuity are a powerful tool in evaluating the approximation order. To evaluate the approximation order through the operatorsS n,3 we use general evaluations, expressed with second-order moduli of continuity, demonstrated in [23]. These give a finer evaluation than the evaluations with the first order modulus. For this we introduce the following notation. Let (X, · X ) be a normed space and D ⊂ X be a compact and convex set. Let e 0 : X → R, e 0 (t) = 1, t ∈ D.
If ϕ ∈ C(X, R) and h > 0, then the usual second-order modulus of a function ϕ ∈ C(X, R) is defined byω With these elements, a particular version of a more general result given in [23] can be expressed in the form: Theorem 5. Let L : C(X, R) → C(X, R) be a positive linear operator. Suppose that X is finite dimensional space with dim X = p. Let x ∈ X. Suppose also that (Lψ)(x) = ψ(x), for all affine functions ψ : X → R.
In [23] an other second-order global continuity modulus is defined: where ϕ ∈ C(D, R), D ⊂ X is a compact and convex set in the normed space X and h > 0. For a more general result given in [23], in a particular case we have the next estimate, which does not depend on the dimension of the space X. Theorem 7. Let L : C(X, R) → C(X, R) be a positive linear operator. Suppose that X is finite dimensional space. Let x ∈ X. Suppose also that (Lψ)(x) = ψ(x), for all affine functions ψ : X → R, for any ϕ ∈ C(D, R), h > 0.
Applying this theorem to operatorsS n,3 we get: Proof. The proof is similar to the proof of Theorem 6.