1. 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
. 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
quadratic splines and showed that the considered split has potential to be used for the construction of
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].
2. Two-Dimensional Schoenberg Type Operators on Arbitrary Nodes
We define two-dimensional Schoenberg type operators as follows.
Let us consider the knot sequence
where
and the knot sequence
where
.
The Greville abscissas associated with knot sequence
are
and the Greville abscissas associated with knot sequence
are
The B-splines
depending on
are defined in the following mode:
and the B-splines
depending on
, by:
Remark 1. If and with , , thenand The following relations are well known:
Analogous relations are fulfilled for .
This notation—
,
,
—is used; i.e.,
Definition 1. Two-dimensional Schoenberg type operator associated with has the formwhere , . Remark 2. By taking into account Remark 1 it follows that if , with and with , then Remark 3. Two-dimensional Schoenberg type operators are linear and positive like one-dimensional Schoenberg operators. This follows immediately from the linearity and from the positivity of one-dimensional Schoenberg operators. Additionally, obviously, is a polynomial of degree at most h in the variable α and with degree at most k in variable β, on each rectangle , with and . Moreover is a B-spline in each variable.
Two-dimensional Schoenberg type operators admit partial continuous derivatives on , since Further, the functions
, defined by
,
,
, for
are considered. The following propositions result directly from Definition 1 and relations (
7).
Proposition 1. For we have
- (i)
;
- (ii)
- (iii)
In the next proposition the notation , is used, and denotes the constant function equal to 1, on the both sets and .
Proposition 2. For we have
- (i)
- (ii)
;
- (iii)
;
- (iv)
.
Theorem 1. For the two-dimensional Schoenberg type operatorsto converge uniformly on to continuous function φ it is sufficient that for any uniformly for , when and Proof. We assume that condition (
10) is fulfilled. Since
is continuous on
, for
,
such that for any
and
with
one has
. Additionally
,
.
Let
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
,
.
Theorem 2. For any , operators given in (9) satisfy inequalitywhere . Proof. Let the continuous function and let . There exist and so that
Let
and
. Then
and
Therefore
. Similarly
. Then
From Remark 2 it results
□
Corollary 1. Ifthen two-dimensional Schoenberg type operatorsconverge uniformly on to φ, for any continuous function φ. 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
is presented.
Theorem 3. The second moment of the two-dimensional Schoenberg type operators for , iswhere with and . Proof. By applying the linearity, Proposition 2(iii) and (iv) follow and the result given in [
18]. □
3. Two-Dimensional Schoenberg Type Operators with Equidistant Knots
Now the case , and equidistant knots is analyzed. More precisely, the equidistant knots are , and the extra-knots are and , respectively , with extra-knots and .
The Greville abscissas are in this case
respectively
The B-splines are
respectively
Two-dimensional Schoenberg type operators, with
and
, with equidistant knots is denoted by
:
and the one-dimensional
k degree Schoenberg operators with equidistant knots are denoted by
.
Lemma 1. The second moment of the two-dimensional Schoenberg type operators , with and , verifies the relationMoreover,for and . Similar relations are true for .
Proof. The exact form of the second moment of the one-dimensional Schoenberg operators was established in [
18]. We have the next cases:
For we have with the maximum ;
For we have , which is an increasing function on with the maximum ;
For we have .
By symmetry, the inequality is also obtained for .
Finally Proposition 2-(iii) can be applied. In the case of Proposition 2-(iv) can be applied. □
The Voronovskaja type theorems are a main topic in studying the convergence properties of the sequences of linear operators. We mention only [
3,
19,
20,
21,
22]. For two-dimensional Schoenberg type operators
, we obtain the following Voronovskaja type theorem.
Theorem 4. The following limit is true:for any , . Proof. From Taylor’s formula, for any
it follows that
with the remainder
where
when
.
Applying operator
in relation (
24) and taking into account Proposition 2 results in
Let . Then there is so that if .
It takes place that
where
, when
.
We have
and
. Then,
From the relation (28) it follows that
From Lemmas 1 and 2 applied in (
29), one has
Since
, for sufficiently great
n we have
. Then, for such
n from Lemma 1 it follows that
. Replacing these in (
26) and taking into account relation (
30), Equation (
23) is immediate. □
Moduli of continuity are a powerful tool in evaluating the approximation order. To evaluate the approximation order through the operators
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
be a normed space and
be a compact and convex set. Let
,
.
If
and
, then the usual second-order modulus of a function
is defined by
With these elements, a particular version of a more general result given in [
23] can be expressed in the form:
Theorem 5. Let be a positive linear operator. Suppose that X is finite dimensional space with dim
. Let . Suppose also thatthenfor any . In the case of operators we get:
Theorem 6. where Proof. We take
,
,
. Therefore
where
The particular case is obtained if is chosen. □
In [
23] an other second-order global continuity modulus is defined:
where
,
is a compact and convex set in the normed space
X and
.
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 be a positive linear operator. Suppose that X is finite dimensional space. Let . Suppose also thatthenfor any . Applying this theorem to operators we get:
Theorem 8. For a function φ continue on and we haveConsequently Proof. The proof is similar to the proof of Theorem 6. □
Remark 4. We can make a comparison between the order of approximation reached by the two-dimensional Schoenberg type operators and that obtained by the two-dimensional Bernstein operators of degree n, which are given bywhere , for . These operators are the most common approximation polynomial operators. For these, the following relation is well known: Thus, if for a function , its values at knots , are known, then applying two-dimensional Bernstein operators we get: On the other hand, if the values of function φ are known at knots , , where (see (15) and (16)), then applying two-dimensional Schoenberg operators given in (19) it results relations (34) and (37). Thus, when using the same tools of measuring for the order of approximations, the advantage is clearly in the favor of Schoenberg operators. Additionally, the volume of computations and the rounding errors are higher in the case of Bernstein operators. Only the smoothness of the image is better than the smoothness of . However, for practical applications, the fact that has continuous partial derivatives of degree 2 offers a sufficient order of smoothness. 4. Conclusions
In this study, a definition of two-dimensional Schoenberg type operators and their properties has been established. The definition was obtained by generalizing the one-dimensional Schoenberg operators’ formula. The exact forms of the second moment of two-dimensional Schoenberg type operators on arbitrary knots, and on equidistant knots, respectively, alongside a Voronovskaja type theorem, are given. The study presented here also contains estimates with moduli of continuity.
The extension of the Schoenberg operators to the two-dimensional case increases significantly the applicability area of the Schoenberg approximation method. The two-dimensional Schoenberg type operators generate sufficient smooth surfaces for practical applications and also offer a very good order of approximation of functions. An important advantage of the definition of two-dimensional Schoenberg type operators established in this study consists of the fact that they have a simple form, and this can lead to their easy application in practice.