Preserving the Shape of Functions by Applying Multidimensional Schoenberg-Type Operators

: The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many ﬁelds of applied sciences. The construction of these operators uses a symmetry regarding the domain of deﬁnition. The degree of approximation by sequences of such operators is given in terms of the ﬁrst and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.


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].
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.

Multidimensional Schoenberg-Type Operators on Arbitrary Nodes
We consider the integers j, m, 1 ≤ j ≤ m; n j > 0; k j > 0; the vector (x 1 , . . . , x m ) ∈ [0, 1] m and the knots sequences ∆ n j ,k j The Greville abscissas associated with division ∆ n j ,k j , 1 ≤ j ≤ m have the next form When The next relations take place and Definition 1. Multidimensional Schoenberg-type operator associated with ∆ has the form where f : [0, 1] m → R, and x = (x 1 , . . . , x m ) ∈ [0, 1] m .

Remark 1.
(i) Symmetrizing the knots ν i j on each components by function σ(x) = 1 − x, x ∈ [0, 1] one obtains also a Schoenberg-type operators of the same degree. If the knots are equidistant, one obtains the same Schoenberg-type operators.
(iv) S ∆ is a polynomial of degree at most k j in each variable x j , (vi) Multidimensional Schoenberg-type operators admit partial continuous derivatives on [0, 1] m , since We consider the next functions: We use the next notations: e 1 (t) = t, t ∈ [0, 1]; e 0 for the constant function equal to 1, on [0, 1] m , 1 ≤ j ≤ m and ∆ j , 1 ≤ j ≤ m denotes the knot sequence use to one-dimensional Schoenberg operators.   (9) to converge uniformly on [0, 1] m to continuous function f , it is sufficient that for any η > 0 Proof. We consider (10) is fulfilled. From f continue function on [0, 1] m , we have ∀ε > 0, ∃η ε > 0 such that for any Let n ε j ∈ N, such that: for n ≥ n ε j . For such n we obtain The norm of the division ∆ is We use the first order modulus of continuity: where Theorem 2. For any f ∈ C([0, 1] m ), operators S ∆ given in (7) satisfy inequality where ϑ = 1 2 max 1≤j≤m {k j + 1}.

Proof.
Let the continuous function f and (x 1 , . . . , Therefore, It results converge uniformly on [0, 1] m to f , for any continuous function f if ∆ → 0.

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 onedimensional Schoenberg operators.
Let k ∈ N. We denote by (S ∆ ϕ)(x) the one-dimensional Schoenberg operators of degree k associated with the knot sequence Using these notations, the following relations are given in [3]: In the following theorems it is considered that n and k are variable.
The convergence is uniform on compact subsets of (0, 1).
We are interested in generalizing these above results in the case of multidimensional Schoenberg-type operators. Let We consider now multidimensional Schoenberg-type operators with equidistant knots on D of the form where f : On D consider the following partial order. If a = (a 1 , . . . , a m ) ∈ D, b = (b 1 , . . . , b m ) ∈ D, we write a ≤ b, iff a i ≤ b i , for 1 ≤ i ≤ m. A function f : D → R is said to be increasing if for any a, b ∈ D, such that a ≤ b, we have f (a) ≤ f (b). Theorem 6. For any integers m ≥ 1, and k ≥ 1, if f : D → R is increasing then S m n,k f is increasing on D, for any n ≥ 1.

Proof. Show that
In order to show (17) it suffices to show that function g(t) = (S m n,k f )(a + tv), t ∈ [0, 1], is increasing, and for this it suffices to have d Because v i ≥ 0, 1 ≤ i ≤ m it suffices to show: Using formula (14) we obtain (17) is true.
In the next two theorems we give generalizations of Theorem 4. We mention that     (18), we obtain (denoting i = i j ): Using formula (15) it follows Consider the moduli of continuity of functions D 2 h and In addition, since f and has continuous partial second derivatives on D, it results that is uniformly continuous and consequently uniformly with regard to indices i 1 , . . . , i j−1 , i j+1 , . . . i m and x ∈ [a, b] m . On the other hand, consider the sequence of one-dimensional Schoenberg operators L n : This sequence of positive linear operators approximates uniformly on [a, b] any function ϕ ∈ C[0, 1]. Moreover, L n e 0 = e 0 , Le 1 = e 1 . Using the well known estimate of Shisha and Mond, we obtain Since lim n→∞ L n e 2 − e 2 = 0 we get uniformly with regard to x j ∈ [a, b] and indices i 1 , . . . , i j−1 , i j+1 , . . . i m . From (22) and (23) we deduce uniformly with regard to x ∈ [a, b] m and indices i 1 , . . . , i j−1 , i j+1 , . . . i m . Taking into account relations (20) and (21) we obtain the uniform limit with regard to x ∈ [a, b] m : Since one obtains the uniform majorization with regard to Finally, it results that (19) is true.
Theorem 9. If f : D → R is strictly convex and has continuous partial second derivatives on D, then for any compact convex set K ⊂ • D there exists an indice n 0 , depending on f and K, such that S m n,k ( f ) is convex for each n ≥ n 0 on K.
Proof. From the hypothesis we obtain the following symmetric positive definite quadratic form: Denote B = {x ∈ R m , x = 1}. Because D is compact and B is compact we obtain that D × B is compact. Because the function F : D × B → R is continuous and strictly positive on the domain of definition, from the Weierstrass theorem one obtains that there exists µ > 0 such that Using Theorems 7 and 8 we obtain for any indices 1 ≤ i, j ≤ m.
Then it results uniformly for x ∈ K and for v ∈ B. Therefore, there exists n 0 ∈ N, such that Inequality (27) says that S m n,k ( f ) is convex on K.

Multidimensional Schoenberg-Type Operators of Degree Three on Equidistant Knots
Let one consider the case with k j = 3; n j = n; the equidistant knots The Greville abscissas are The B-splines are with 1 ≤ j ≤ m. Multidimensional Schoenberg-type operators with equidistant knots, denoted in the sequel by S m n,3 , for k j = 3, n j = n, 1 ≤ j ≤ m, are: In this section we present certain special results for the cubic splines, which can be proved analogously as in [11].
for x j ∈ 2 n , n−2 n and 1 ≤ j ≤ m.
Using Lemma and the inequality given in [21]: where S n,k 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 f ∈ C 2 ([0, 1] m ), (x 1 , . . . , x m ) ∈ (0, 1) m .
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 f ∈ C(D), h > 0 One obtains: where f ∈ C([0, 1] m ), h > 0, (x 1 , . . . , x m ) ∈ [0, 1] m , n ∈ N, n ≥ 5. Consequently: 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:

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.