1. Introduction
The theory of splines approximation was founded by Schoenberg and became one of the main chapters of approximation theory. Now there is a vast literature dedicated to spline approximation. We refer the reader to the monograph of Schumaker [
1] for historical notes. The success of this type of approximation is due both to the nice mathematical theory and to the great efficiency in practical applications. In practice, the spline approximation is more efficient then the polynomial approximation.
In [
2], Schoenberg considered also a particular method of approximation of functions by splines, with the aid of certain positive linear operators, which are named the Schoenberg operators. Important contributions in the study of these operators are due to Marsden [
3].
In more recent times, the topic of one-dimensional Schoenberg spline operators are presented in papers written for instance in Gonska [
4,
5,
6,
7], Tachev [
4,
5,
6,
7,
8], Beutel [
5,
6,
7,
9], Kacsó [
4,
5,
6,
7] and our papers [
10,
11,
12].
The subject of multidimensional spline is developed in many papers. We can specify here the paper [
13] where the approximation of functions using multivariate splines is presented and the monograph [
14], which is dedicated to the theory of multivariate splines. We mention also the paper [
15] where the multivariate polynomial interpolation is approached, the paper [
16] where a computationally effective way to construct stable bases on general non-degenerate lattices is presented, Reference [
17] where the subject of Hermite-vector splines and multi-wavelets is developed and the paper [
18] in which a generalization of bases, namely B-spline frames, is approached. Estimates of approximation by linear operators in the multidimensional case are established in [
19].
As exemplification of the application of the Schoenberg operators in practice we mention the recent paper [
20] where one-dimensional Schoenberg spline operators were used, obtaining a substantial improvement of the clear sky models which estimate the direct solar irradiance.
The present paper is a continuation of paper [
12], where two-dimensional Schoenberg operators were considered. Now we extend this definition in multidimensional case and we establish certain properties of them.
Several important connections with symmetry exist in this study. Because these operators present a symmetry in their construction, the computation of their moments is made by symmetry. The symmetry is also used in establishing the estimates with second order moduli, which are defined with the aid of finite symmetric differences. On the other hand, we study the property of preservation of convexity and this property can be described using the Hessian of functions, which is a symmetric quadratic form.
2. Multidimensional Schoenberg-Type Operators on Arbitrary Nodes
We consider the integers
,
;
; the vector
and the knots sequences
The Greville abscissas associated with division
,
have the next form
The B-splines
depending on
are
When
with
,
we have:
The next relations take place
and
where
.
We consider
Definition 1. Multidimensional Schoenberg-type operator associated with has the formwhere , and . Remark 1. (i) Symmetrizing the knots on each components by function , one obtains also a Schoenberg-type operators of the same degree. If the knots are equidistant, one obtains the same Schoenberg-type operators.
- (ii)
For , with and , then - (iii)
Multidimensional Schoenberg-type operators are linear and positive.
- (iv)
is a polynomial of degree at most in each variable , , on each domain , with and .
- (v)
is a B-spline in each variable.
- (vi)
Multidimensional Schoenberg-type operators admit partial continuous derivatives on , sincewhere , .
We consider the next functions: , and , , for , .
Proposition 1. For we have
- (i)
;
- (ii)
;
- (iii)
We use the next notations: , ; for the constant function equal to 1, on , and denotes the knot sequence use to one-dimensional Schoenberg operators.
Proposition 2. For we have
- (i)
,
- (ii)
- (iii)
, .
Theorem 1. For multidimensional Schoenberg-type operatorsto converge uniformly onto continuous function f, it is sufficient that for anyuniformly for, when,
. Proof. We consider (10) is fulfilled.
From f continue function on , we have , such that for any and with it results .
Also
Let
, such that:
for
. For such
n we obtain
□
The norm of the division
is
where
.
We use the first order modulus of continuity:
where
,
.
Theorem 2. For any , operators given in (7) satisfy inequalitywhere . Proof. Let the continuous function f and . For any there is , such that . Then , for and .
Let
. Then
and
Then, for
we have
□
Corollary 1. Multidimensional Schoenberg-type operatorsconverge uniformly on to f, for any continuous function f if . 3. Preservation of Monotonicity and Convexity by Multidimensional Schoenberg-Type Operators with Equidistant Knots
In this section, we will extend some results obtained by Marsden in the case of one-dimensional Schoenberg operators.
Let
. We denote by
the one-dimensional Schoenberg operators of degree
k associated with the knot sequence
, where
, and the Greville abscissas
, (
,
). The B-spline of degree
k associated to
is denoted by
,
. Next, denote
. The corresponding B-splines of degree
associated with the knot sequence
by
,
and the corresponding Greville abscissas is denoted by
. In addition, for
, denote
and the corresponding B-splines of degree
associated with the knot sequence
by
,
. Using these notations, the following relations are given in [
3]:
In the following theorems it is considered that n and k are variable.
Theorem 3 ([
3])
. Let and . Then:- (i)
uniformly on ;
- (ii)
uniformly on .
Theorem 4 ([
3])
. Let and the interior knots of Δ. Let . Then The convergence is uniform on compact subsets of . Theorem 5 ([
3])
. Let and . Then:- (i)
If on then on ;
- (ii)
If on then on .
We are interested in generalizing these above results in the case of multidimensional Schoenberg-type operators.
Let an integer . Denote .
We consider now multidimensional Schoenberg-type operators with equidistant knots on
D of the form
where
,
,
and
On D consider the following partial order. If , , we write , iff , for . A function is said to be increasing if for any , such that , we have .
Theorem 6. For any integers , and , if is increasing then is increasing on D, for any .
Proof. Let , . Let . Write . Then , . In order to show (17) it suffices to show that function , , is increasing, and for this it suffices to have , for .
Because
,
it suffices to show:
Denoting
,
one obtains
Fix the indices
and define function
, given by
Using formula (14) we obtain
However, , since f is increasing. In addition, taking into account that , and , , , , relation (17) is true. □
In the next two theorems we give generalizations of Theorem 4. We mention that means the interior of the set D.
Theorem 7. Let . Let . For , if f admits the continuous derivatives on D, then for any compact set we have Proof. It suffices to consider only compacts of the form
, where
, because for any compact
there exists an interval
, such that
. Using the function
h given in (18), we obtain (denoting
):
Using formula (15) it follows
where
.
For
, we have
The support of a function
is the interval
. Then, if
and
, and
, then
, for
, and
. In addition, for
we obtain
, so that we can write more simply
Consider the moduli of continuity of functions
and
:
where
. Because
is a restriction of function
we obtain
In addition, since
f and has continuous partial second derivatives on
D, it results that function
is uniformly continuous and consequently
We have
since
, for
. It follows
uniformly with regard to indices
and
. On the other hand, consider the sequence of one-dimensional Schoenberg operators
This sequence of positive linear operators approximates uniformly on
any function
. Moreover,
,
. Using the well known estimate of Shisha and Mond, we obtain
Since
we get
uniformly with regard to
and indices
. From (22) and (23) we deduce
uniformly with regard to
and indices
. Taking into account relations (20) and (21) we obtain the uniform limit with regard to
:
Now consider the
- dimensional Schoenberg operator
, given by
where
and
With the choice
, for fixed
and
and using Theorem 2 we obtain, for fixed
:
Since
one obtains the uniform majorization with regard to
:
Finally, it results that (19) is true. □
Theorem 8. Let and . For indices if f admits the continuous second derivative on D, then we have Proof. For
we find
where we denoted by
g, the function
and
and
. Using formula (14) two times, it follows that we can write
We have the limit
uniformly with regard to index
,
and
. It follows that the limit
is uniform with regard to
,
and
.
Using the property of uniform approximation of two-dimensional Schoenberg operators
the Shisha and Mond estimate and the inequality
, for
, we get, like in Theorem 7
and this limit is uniform with regard to
and indices
,
,
. It follows that
and the limit is uniform with regard to
and indices
,
,
. Then we have the limit
uniform with regard to
. Finally, we apply the property of uniform approximation of the Schoenberg operators of degree
:
to the function
with
and
fixed. One obtains relation (24). □
Theorem 9. If is strictly convex and has continuous partial second derivatives on D, then for any compact convex set there exists an indice , depending on f and K, such that is convex for each on K.
Proof. From the hypothesis we obtain the following symmetric positive definite quadratic form:
Denote
. Because
D is compact and
B is compact we obtain that
is compact. Because the function
is continuous and strictly positive on the domain of definition, from the Weierstrass theorem one obtains that there exists
such that
Using Theorems 7 and 8 we obtain
for any indices
.
Then it results
uniformly for
and for
. Therefore, there exists
, such that
Inequality (27) says that is convex on K. □
4. Multidimensional Schoenberg-Type Operators of Degree Three on Equidistant Knots
Let one consider the case with ; ; the equidistant knots ; the extra-knots and .
The Greville abscissas are
with
.
The B-splines are
with
.
Multidimensional Schoenberg-type operators with equidistant knots, denoted in the sequel by
, for
, are:
In this section we present certain special results for the cubic splines, which can be proved analogously as in [
11].
Lemma 1. The second moment of the multidimensional Schoenberg-type operators , with and , verifies the relations Moreover,for and . Using Lemma and the inequality given in [
21]:
where
denotes the Schoenberg one-dimensional operator of order
k with equidistant knots, one obtains:
From Lemma 1 and Lemma 2 and the fact that Schoenberg preserves linear functions, one can deduce the following Voronovskaja-type result, in a similar mode as in [
11].
Theorem 10. The following limit is true:for any , . Because Schoenberg preserves linear functions there exists the possibility of expressing the degree of approximation in a more refined mode, using second order moduli of continuity. The following estimates can be obtained similarly to [
11] by applying certain general estimates with moduli of continuity proved in [
19].
Firstly, consider the usual second order modulus
where
,
One obtains:
Theorem 11. where A global second modulus of continuity can be defined by:
Using this modulus one can obtain an estimate which is independent on the dimension m:
Theorem 12. Let the function f continue on and . We have Remark 2. The usual polynomial operators used in approximation have an approximation order only of the type .
5. Conclusions
The Schoenberg operators are practical tools to approximate functions, knowing the values of them in a finite number of points. Schoenberg operators attach to a function a particular type of spline of a freely chosen degree. It is not necessary to use high-degree splines in order to obtain a desired approximation order. It is usually sufficient to use 3rd order splines. This makes the calculation volume substantially lower than in the case of polynomial approximation.
In approximation of functions, the degree of approximation is not the unique objective. The preservation of certain shape properties of functions is also worth studying. Among these supplementary preservation proprieties, two special types are usually studied: the possibility of simultaneous approximation of functions and of their derivatives of different orders and the preservation of convexity of different orders, including the monotonicity and the usual convexity. These types of properties are known to be true for the one-dimensional case of Schoenberg operators. We put in evidence that they are true in great measure for the multidimensional case. It is well known also that maybe the more important polynomial approximation operators, namely the Bernstein operators, have very good properties for preserving different behaviors of functions. In fact, it is natural that the Schoenberg operators, which can be regarded as generalizations of Bernstein operators, maintain at least in part these good properties. On the other hand, by taking into account that Schoenberg operators offer a great improvement of the order of approximation for the same order of computation, they turn out to be a very powerful tool in the theory of function approximation. In this direction, other properties of preserving certain classes of functions or the simultaneous approximation can be taken into account for further studies.
The results obtained in this paper are connected to the notion of symmetry in several aspects, namely, in the construction of operators, in using the symmetric tools in estimates and in property of convexity, which is given using symmetrical expressions. We believe that this paper can offer a useful tool to specialists with concerns in many areas of practical approximation.