© Hindawi Publishing Corp. SHAPE-PRESERVING MULTIVARIATE POLYNOMIAL APPROXIMATION IN C[−1,1] m

We construct multivariate polynomials attached to a function f of m variables, m ≥ 2 
, which approximate f with Jackson-type rate involving a multivariate Ditzian-Totik ω 2 φ -modulus and preserve some natural kinds of multivariate monotonicity and convexity of function. The result extends the bivariate case in Gal (2002), but does not follow straight from it and requires a careful calculation.


Introduction.
In a very recent paper [2] the following approximation result for the bivariate case was proved.
By using the convexity of order (n 1 ,...,n m ) and the Ditzian-Totik modulus of smoothness ω ϕ 2 (f ; δ 1 ,...,δ m ) for functions of m variables defined on [−1, 1] m , in this paper we extend Theorem 1.1 to the case of functions of m variables, m > 2.
A simple comparison between the statements of Theorem 1.1(ii), (iv) and Theorem 3.1(ii), (iv), shows that Theorem 3.1 cannot be trivially suggested (by mathematical induction) from Theorem 1.1, that is, we can say that the m = 2 case is not representative for the general case m ∈ N (as, e.g., would be the m = 3 case).Also, the proof of Theorem 3.1 requires more intricate calculation than in the case of Theorem 1.1, which motivates us in Section 3 to prove the main result in its full generalization.

2.
Preliminaries.This section contains the concepts of convexity and of modulus of smoothness for multivariate functions, which will be used in Section 3.
Following the ideas in [6, page 78], we can introduce the following definition.
Definition 2.1.The function f : where the above symbol •] is considered with respect to the x i variable, respectively, for all i = 1,m).

Remark.
(1) For m = 2, we get the concept in [6, page 78 then by the mean value theorem, it follows that the condition implies that f is convex of order (n 1 ,...,n m ).
The method in [1, Chapter 12] suggests introducing the following definition. where where C 2 k denotes the usual binomial coefficients.

Main result.
Our main result can be stated as follows.
Proof.If g : [−1, 1] → R, then, according to [4, relation (5)], the approximation polynomials are given by where , and R j,n (x) are suitable chosen polynomials of degree less than or equal to n.
According to [4, Theorem 1], we have where ω ϕ 2 (g; δ) is the usual Ditzian-Totik modulus of smoothness and C > 0 is independent of g and n.
We will construct the polynomials P n 1 ,...,nm (f )(x 1 ,...,x m ) by applying the tensorproduct method (see, e.g., [5, pages 195-296]).We obtain (by mathematical induction) ..,m, are constructed as in the univariate case in [4].The value of f on the point (−1,...,−1) by definition can be represented as a divided difference by  and ξ (j) i j ,n j , ξ (j) i j +1,n j , respectively, and so on.Note that, finally, that is, it is a divided difference with m lines, having two nodes on each line.
For any univariate function g, we have that is, passing to supremum with g ≤ 1, for the linear operator P n , we obtain where c > 0 is independent of n.Now applying [3, Theorem 5], we immediately get where ω ϕ 2,x i (f ; δ i ), i = 1,m, are the partial moduli of smoothness defined by where In what follows we will prove the shape-preserving properties.(i) Suppose f is convex of order (0,...,0).According to Remark (2) in Section 2, we have to prove that We get ..,m ≥ 0 (f is convex of order (0,...,0)), and from the univariate case, each (ii) By hypothesis on f , it follows that all the quantities and [•; f ] 1,...,m in the expression of P n 1 ,...,nm (f ) are greater than or equal to 0.
By R i k ,n k (−1) = 0, for all i k = 0,n k , k = 1,m, we immediately get Also, from the univariate case, we have The above hypothesis and simple calculations (similar to those in the bivariate case, see the proof of [2, Theorem 3.1(ii), pages 30-31]) immediately imply that P n 1 ,...,nm (f )(x 1 ,...,x m ) is convex of order (s 1 ,...,s m ), which proves (ii).
(iii) We have to prove that Applying with respect to each variable the relation in the univariate case (see [4, (5), page 473]), that is, where we get where F(x 1 ,...,x m ) is a sum of several expressions of the form (1 with distinct indexes i 1 ,...,i k , k < m, C a real constant (which can be different at each occurrence) and E(x 1 ,...,x m ) is a sum of several expressions, each expression being a simple or multiple sum of terms, where each term is represented by product between: (a) various R i k ,n k (x k ), or a product of distinct R i k ,n k (x k ) (having at most m−1 terms in that product), (b) a divided difference of f on one, two, or three nodes with respect to each variable x k , such that at least with respect to one variable the divided difference is taken on three nodes, (c) a positive quantity of the form (ξ or the product of such distinct quantities.Moreover, the above-mentioned expressions in E(x 1 ,...,x m ) which depend on the variables x k , k = 1,m, through R i k ,n k (x k ), are of two kinds: (1) expressions which do not depend on all variables x k , k = 1,m; (2) expressions which depend on all variables x k , k = 1,m, but at least one R i k ,n k (x k ) = 1 + x k .We exemplify the passing from m = 2 to m = 3.Therefore, let f be a function of three variables, that is, f = f (x 1 ,x 2 ,x 3 ).
Applying the formula in the univariate case (specified at the beginning of (iii)) with respect to the variables x 1 and x 2 , and by the formulas in [2, pages 31-32], we immediately get where all the divided differences are considered with respect to the variables x 1 , x 2 , and x 3 is arbitrarily fixed.Also, recall that the formula for S * 0,0 (f )(x 3 ) is given by [2, page 31] and depends on the values of f (•, •,x 3 ) on some nodes, where f is considered as a function of the variables x 1 and x 2 .Now, applying the formula in the univariate case with respect to x 3 to P n 1 ,n 2 (f )(x 1 , x 2 ,x 3 ), that is, to each term of it, and taking into account the recurrence formula satisfied by the divided differences, we immediately obtain P n 1 ,n 2 ,n 3 (f )(x 1 ,x 2 ,x 3 ) of the claimed form.
As a conclusion, all these immediately imply that i by the hypothesis on f and by the conditions R i k ,n k (x k ) ≥ 0, for all i k = 0,n k , x k ∈ [−1, 1], k = 1,m (see [4]).
(iv) Firstly, we recall that by construction we have (see [4]) We have to check the inequalities for all r ∈ {2,...,m}, p ∈ {1,...,m}, i k = i j , if i = j, r = r 1 +•••+r p , where at least one r l is equal to 2 and r k ∈ {0, 1, 2}, k = 1,p.By hypothesis, the divided differences of f which contains, at least on a line, three distinct points, all are greater than or equal to 0. Then, taking into account the forms of F(x 1 ,...,x m ) and E(x 1 ,...,x m ) described at the previous point (iii), we immediately obtain the required conclusion.(2) Since in the univariate case (i.e., m = 1) the property in Theorem 3.1(i) reduces to the usual increasing monotonicity and in this case by [7] we know that ω 2 ϕ (f ; •) cannot be replaced by higher-order moduli of smoothness ω k ϕ (f ; •) with k ≥ 3, it follows that for arbitrary m ≥ 2, the same phenomenon is expected.

. 5 )
where the notation in the right-hand side denotes the divided difference of f with m lines (see Definition 2.1) with −1 on each line.Then [•; f ] k denotes the divided difference obtained from the above by replacing the line k (which has only one node, −1) with another one composed of the two nodes ξ (k) i k ,n k and ξ (k) i k +1,n k (the rest of lines remain unchanged), also [•; f ] k,j , k = j, denotes the divided difference obtained (from the same divided difference which represents f (−1,...,−1)) by replacing the lines k and j (which have only the node −1) by lines composed of the pairs of nodes ξ