Blending Function Interpolation: A Survey and Some New Results

Interpolation operators on a triangle with one curved side

“…First, we suppose that the function F :T h → R has the partial derivatives F (1,0) and F (0,1) on the side Γ 3 .…”

“…We consider the Birkhoff-type operators B 1 and B 2 defined respectively by (B 1 F )(x, y) = F (0, y) + xF (1,0) (g(y), y)…”

“…Suppose that the real valued function F is defined on the triangleT h and it possesses the partial derivatives F (1,0) and F (0,1) on the side Γ 3 . We consider the operators H 1 and H 2 defined by…”

“…The idea comes from the paper of Barnhill and Gregory [3], where there is considered a triangle with one curved side and there are used Lagrange projectors on the straight sides, respectively, Taylor projector on the curved side. Such operators were also studied in many other papers in connexion with their applications in computer aided geometric design (see, e.g., [1,2,4,5]) and in finite element analysis (see, e.g., [1,8,16,[18][19][20]29]). In [11] we have obtained operators on a tetrahedron with three curved sides.…”

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

“…Gordon, there have been constructed interpolation operators of Lagrange, Hermite and Birkhoff type, that interpolate the values of a given function or the values of the function and of certain of its derivatives on the boundary of a triangle with straight sides. These operators were applied in computer aided geometric design (see, e.g., [1]- [3], [5]]) and in finite element analysis (see, e.g., [1], [7], [8], [13], [17]- [19], [21], [22]). …”