BERNSTEIN-TYPE OPERATORS ON TRIANGLES

. The aim of the paper is to construct some univariate Bernstein-type operators on triangle, their product and Boolean sum, which interpolate a given function on the edges respectively at the vertices of triangle. Using the modulus of continuity and the Peano’s theorem the remainders of corresponding approximation formulas are studied.

The aim of this paper is to construct Bernstein-type operators wich also interpolate the value of a given function on the border of triangle.Using modulus of continuity respectively Peano's theorem the remainders of the corresponding approximation formulas are as well studied.The accuracy of approximation is also illustrated by graphics of given functions and of the suitable Bernsteintype approximation.
By affine invariance it is sufficient to consider only the standard triangle T h = (x, y) ∈ R 2 | x 0, y 0, x + y h , for h > 0.

UNIVARIATE OPERATORS
Let f be a real-valued function defined on T h .Through the point (x, y) ∈ T h , one considers the parallel lines to the coordinate axes which intersect the edges Γ i , i = 1, 2, 3, of the triangle at the points (0, y) and (h − y, y) respectively (x, 0) and (x, h − x) (Figure 1).
One considers the Bernstein-type operators B x m and B y n defined by where where e ij (x, y) = x i y j and dex (B x m ) is the degree of exactness of the operator B x m .
Proof.The interpolation property (i) follows from the relations and Regarding the properties (ii), we have Remark 1.In the same way it is proved that: Now, let us consider the approximation formula where ω (f ( , y) ; δ) is the modulus of continuity of the function f with regard to the variable x.

PRODUCT OPERATORS
Let P mn = B x m B y n respectively Q nm = B y n B x m be the products of operators B x m and B y n .We have Remark 4. The nodes of the operator P mn are as in the Figure 2, for i = 0, m, j = 0, n, and y ∈ [0, h].Theorem 4. The operator P mn satisfies the following relations: The proofs follow by a straightforward computation.The property (i) or (ii) imply that (P mn f ) (0, 0) = f (0, 0).Remark 5.The product operator P mn interpolates the function f at the vertex (0, 0) and on the hypothenuse x + y = h of the triangle T h .
The product operator Q nm , given by has the nodes as in Figure 3, for i = 0, m, j = 0, n, x ∈ [0, h], and the Fig. 3. Nodes of operator Qnm. properties Let us consider the approximation formula Proof.We have After some transformations, one obtains It follows, Taking into account that

BOOLEAN SUM OPERATORS
Let be the Boolean sums of the Bernstein-type operators B x m and B y n .
Theorem 6.If f is a real-valued function defined on T h then Proof.We have The interpolation properties of B x m , B y n and the properties (i)-(iii) of the operator P mn imply that Let R S mn f be the remainder of the Boolean sum approximation formula f = S mn f + R S mn f. 2), ( 7) and (10), the proof follows.
Remark 6. Analoguous relations can be obtained for the remainders of the product approximation formula

EXAMPLES
Finally, one considers two test functions, generally used in literature (see, e.g.[22])

2 •
In Figure4we plot the graphs of f 1 and f 2 .