BIERMANN INTERPOLATION OF BIRKHOFF TYPE

. If P 0 , P 1 , ..., P r and Q 0 , Q 1 , ..., Q r are Birkhoﬀ univariate projectors which form the chains

Let X, Y be the linear space on R or C. The linear operator P defined on space X is called projector if P 2 = P .The operator P c = I − P , where I is identity operator, is called the remainder projector of P .If P is projector on space X then the range space of projector P is denoted by ( 1) The set of interpolation points of projector P is denoted by P(P ).
If P 1 and P 2 are projectors on space X, we define relation "≤" by Let f ∈ C(X × Y ) and x ∈ X.We define f x ∈ C(Y ) by For y ∈ Y we define y f ∈ C(X) by y f (s) = f (s, y), s ∈ X.
Let P be a linear and bounded operator on C(X).The parametric extension P of P is defined by (3) (P f )(x, y) = (P y f )(x).
If P is a linear and bounded operator on C(Y ) then the parametric extension Proposition 2. Let r ∈ N, P 0 , P 1 , . . ., P r be univariate interpolation projectors on C(X) and Q 0 , Q 1 , . . ., Q r univariate interpolation projectors on C(Y ).Let P 0 , . . ., P r , Q 0 , . . ., Q r be the corresponding parametric extension.We assume that (5) is projector and it has representation Moreover, we have , where P c = I − P , I the identity operator.
For the proof of Proposition 2 one can see [6].Remark 3. If P i , i = 0, r, and Q j , j = 0, r, are Lagrange interpolation projectors which form the chains with respect to relation "≤", the projector B r is called Biermann interpolation projector (see [6]).In [10] we instead the Lagrange projectors by Hermite projectors.In this article, our objective is to construct chains of Birkhoff interpolation projectors and, with their aid, the Biermann interpolant of Birkhoff type.
Moreover, we have The tensor product projector P m Q n of bivariate interpolation has representation and it has the interpolation properties The projectors P 0 , . . .
called Biermann projector of Birkhoff type and which has the interpolation properties The properties (15) and (16) result using Proposition 1.

Let be
According to [5], the cardinal functions g i,m , i = 0, m, are given by recurrence relations for m = 0, r.
One remark that the cardinal functions g i,m depend only the nodes x 0 , x 1 , . . ., x i−1 .It follows that the functions g i,m , i = 0, r, are the same for m = i, r.One denotes We have analogous relations for the cardinal functions g j,n and we denote g j (x) := g j,m (x), j = 0, r, m = j, r.

It follows that relations (11) hold.
We can define the Biermann operator of Abel-Goncharov type (18) where P i , i = 0, r, and Q j , j = 0, r, are the parametric extensions.
Taking into account (7) we obtain the following representation for Biermann interpolant where Using Proposition 1, the Biermann interpolant B AG r has the interpolation properties Remark 5.The set of nodes form a triungular grid (x 0 , y r ) (x 0 , y r−1 ), (x 1 , y r−1 ) . . .
and relation (8), the approximation order of Biermann interpolant of Abel-Goncharov type is r + 1, i.e., Remark 6.The approximation order of Biermann operator B AG r is the same with the approximation order of tensor product operator P r Q r .If P is an interpolation projector we denote by I P (f ) the set of data about f (values of function f and/or certain of its partial derivatives at nodes).We have and It follows that the Biermann operator B AG r is more efficient than tensor product operator P r Q r .
If we denote h = x 1 −x 0 = y 1 −y 0 , from [1] we can obtain cardinal functions by relations where the functions ∆ i , i = 0, r, are given by recurrence relations One remark that the cardinal function l m 0,i and l m 1,i don't depend of m.We can make notation We have analogous relations for cardinal functions l n 0,j , l n 1,j and we denote l 0,j (y) := l n 0,j (y), j = 0, r, n = j, r, l 1,j (y) := l n 1,j (y), j = 0, r, n = j, r.
The indices sets of derivatives verify relations (11) because We can define the Biermann operator of Lidstone type where P i , i = 0, r, and Q j , j = 0, r, are the parametric extensions.
From [1] we have Taking into account (8), the approximation order of Biermann operator B L r is 2r, i.e.Remark 7. The approximation order of Biermann operator B L r is the same with the approximation order of tensor product operator P r Q r .We have where I P (f ) is the set of data about f used by interpolation projector P .It follows that the Biermann operator B B r is more efficient than tensor product operator P r Q r .