Extension of Some Positive and Linear Operators to Domains with Curved Sides

There are constructed some Cheney-Sharma type operators defined on a square with one curved side. They are extensions of the Cheney-Sharma type operators of second kind, given by E.W. Cheney and A. Sharma in [14], to the case of a curved sided domain. There are constructed the univariate Cheney-Sharma type operators, their product and boolean sum operators and there are studied their properties, their orders of accuracy and the remainders of the corresponding approximation formulas. Finally, there are given some illustrative examples.


Introduction
The aim of this paper is to construct some Cheney-Sharma type operators that have some interpolatory properties on a curved sided domain.There will be studied the interpolation properties, the orders of accuracy and the remainders of the corresponding approximation formulas.
Using the interpolation properties of such operators, blending function interpolants can be constructed, that exactly match function on some sides of a rectangular region.Important applications of these blending functions are in finite element method for differential equations problems with Dirichlet boundary conditions or for construction of surfaces which satisfy some given conditions.
In order to match all the boundary information on a curved domain (as in Dirichlet, Neumann or Robin boundary conditions for differential equation problems), there were considered interpolation operators on domains with curved sides (see, e.g., [4], [6], [9]- [12], [15], [16], [20], [21], [23]).Approximation operators on polygonal domains with some curved sides have also important applications especially in finite element method for differential equations with given boundary conditions and in the piecewise generation of surfaces in computer aided geometric design.
2) In [26], it has been proved that the Cheney-Sharma operator m Q interpolates a given function at the endpoints of the interval.
3) In [14] and [26], it has been proved that the Cheney-Sharma operator Q m reproduces the constant and the linear functions, so its degree of exactness is 1 (denoted dex(Q m )=1).For hR + , let D h be the square with one curved side having the vertices V 1 =(0,0), V 2 =(h,0), V 3 =(h,h) and V 4 =(0,h), three straight sides, 1 For m,n N, b, E R + , we consider the following extensions of the Cheney-Sharma operator given in (1):  Figure 3.
In Figures 2 and 3 (ii) The proof follows the same steps as for (i).), (

Theorem. The operators
Proof.The proof follows by a straightforward computation.

Boolean sum operators
We consider the Boolean sums of the operators generally used in the literature, (see, e.g., [24]).In Figure 6

Conclusions
The univariate operators obtained here, x m Q and , y n Q interpolate a function F:D h →R on two sides of the domain D h , their products, 1 mn P and 2 nm P , interpolate on the vertices of D h , and their boolean sum operators interpolate on the entire frontier of D h .There are generated the interpolation formulas and there are given the expressions of the corresponding remainders using Peano's theorem.The good approximation properties could be seen in the figures and in the table with errors presented in the previous section.Using the interpolation properties of these operators, blending function interpolants, that exactly match function on some sides of a curved domain region, can be constructed.Important applications of these blending functions are in finite element method for differential equations problems with Dirichlet boundary conditions or for construction of surfaces which satisfy some given conditions.

Figure 1 .
Figure 1.The square D h .

E
Remark.As Cheney-Sharma operator of second kind interpolates a given function at the endpoints of the interval, we may use the operators x m Q and y n Q as interpolation operators on D h .
The proof follows by the property dex(Q m )=1.We consider the approximation formula , given in Figures4 and 5 .

Figure 6 .
Figure 6.The graphs of F, , F Q x m

Table 1 .
The maximum approximation errors.