ON SOME BIVARIATE INTERPOLATION PROCEDURES

. In an important paper published in 1966 by the ﬁrst author [10] was introduced and investigated a very general interpolation formula for univariate functions, which includes, as special cases, the classical interpolation formulae of Lagrange, Newton, Taylor and Hermite. The purpose of the present paper is to extend that formula to the two-dimensional case. The remainders are expressed by means of partial divided diﬀerences and derivatives.


INTRODUCTION
In this paper we start from a general decomposition formula of divided differences defined for a function f ∈ C(D), D = [a, b] × [c, d], and for some groups of nodes from the rectangle D. We deduce a general interpolation formula for bivariate functions, corresponding to some general arrays of points.As special cases we obtain several classical types of interpolation polynomials, including Lagrange, Newton, Biermann, Taylor and Hermite.

PRELIMINARIES
We first recall some of the principal results obtained in the paper [10].Then we shall start from an array of M + 1 = p 0 + p 1 + • • • + p m + m + 1 points containing m + 1 groups of nodes, denoted, by using subscripts and superscripts, by (a k i ) i = 0, . . ., p k , k = 0, . . ., m.Let us use the following explicit notation for this array .
We assume that a ≤ a k 0 < a k 1 < • • • < a k p k ≤ b, k = 0, . . ., m.The key role in deducing a general interpolation formula corresponding to the function f ∈ C[a, b] and the points (a k i ), i = 0, . . ., p k , k = 0, . . ., m, is the following decomposition formula for divided differences on the distinct nodes where and The above brackets represent the symbol for divided differences.If we introduce the node polynomial then we can write: By using the decomposition formula (2.2) we can obtain the Stancu general interpolation formula where, in terms of divided differences, we have The remainder of formula (2.4) has the following expression where ] is the divided difference on all the points from the table A and x.Now let us present three remarkable special cases of the above approximation formula.
(i) If p 0 = p 1 = • • • = p m = 0 then we have a single column in the array (2.1) and the Stancu approximation formula (2.4) reduces to the Lagrange interpolation formula corresponding to the function f and the nodes a 0 0 , a 1 0 , . . ., a m 0 .(ii) In the case m = 0 then the array A reduces to the nodes from the first row and we obtain the Newton interpolation formula corresponding to the nodes a 0 0 , a 0 1 , . . ., a 0 p 0 and the function f .
(iii) If we assume that the nodes from the group a k 0 , a k 1 , . . ., a k p k tend to the same value b k , k = 0, . . ., m, then the polynomial (2.5) becomes the Hermite osculatory interpolation polynomial under the form given in 1931 by P. Johansen [3]: where we use the notations It should be noticed that this polynomial can be written also under the more explicit form, given in 1948, by W. Simonsen [5]: where the basic osculatory interpolation polynomials h k,j (x) satisfy the relations h

1) leads us to the following triangular array of m(m + 1)/2 base points
and in the Stancu interpolation polynomial (2.5) we have to replace

SOME BIVARIATE INTERPOLATION FORMULAS
Let C(D) be the space of all real-valued functions continuous on the rec- r = 0, . . ., n.We denote by B the table of these points .
We want to approximate the function f ∈ C(D) by an interpolation polynomial (T f )(x, y)(x, y) relative to the grid of nodes from D: ), i = 0, . . ., p k , j = 0, . . ., s r , k = 0, . . ., m, r = 0, . . ., n}.In order to find the expression of this bivariate interpolation polynomial using these nodes, we first apply formula presented at (2.4), with respect to the first variable and we obtain (t) .The remainder has the following expression (Rf )(x; y) = u(x)[x, x 0 0 , . . ., a 0 p 0 ; . . .; a m 0 , . . ., a m pm ; f (t, y)], where u(x) = γ 0 (x)γ 1 (x) . . .γ m (x) = u k (x)γ k (x).Then we apply the above result with respect to the variable y and the points b r j from the array B given at (3.1).We obtain On the other hand we used the bidimensional divided difference .
The remainder of the interpolation formula (3.2) has the following expression

INTERPOLATION FORMULAS USING A RECTANGULAR OR A TRIANGULAR GRID OF NODES
A) In the special cases 0 in the tables (2.1) and (3.1) remain only the first columns (a k 0 ), k = 0, . . ., m, and (b r 0 ), r = 0, . . ., n and the nodes will be the points M k,r (a k 0 , b r 0 ) which are at the intersections of the vertical lines x = a k 0 , k = 0, . . ., m, with the horizontal lines y = b r 0 , r = 0, . . ., n, in the plane.In this case the interpolation polynomial (3.3) becomes where The corresponding remainder of the interpolation formula can be expressed by means of the nodal polynomials and the partial divided differences, namely If we now assume that the function f has continuous partial derivatives f (p,q) (x, y) on the rectangle D then this remainder can be expressed in the following form where we have and is the bidimensional divided difference of the function f on the indicated nodes.The remainder of the interpolation formula (4.5) has the following expression, in terms of partial divided differences, (R p 0 ,s 0 f )(x, y) =u p 0 (x)[x, a 0 0 , a 0 1 , . . ., a 0 p 0 ; f (t, y)] If f ∈ C p 0 ,s 0 (D) then we can obtain the following estimation for this remainder C) If we use the notations p 0 = p, a 0 i = x i , b 0 j = y j and assume that s 0 = p − i, i = 0, . . ., m; j = 0, . . ., n, then we arrive from (4.6) at the Biermann interpolation polynomial [1], [9]: where x 0 , x 1 , . . ., x i y 0 , y 1 , . . ., y j ; f (t, z) .
The Biermann polynomial is of total (global) degree p in x and y and uses a triangular array of base nodes (x i , y i ), i = 0, . . ., p, j = 0, . . ., (p − i).
By using the integration by parts, the first author has obtained in [8] the following integral representation for the remainder of the Taylor-type formula (4.10) (Rf )(x, y) = It should be further noted that employing the Biermann interpolation polynomial given at (4.8) we can obtain as a limit case the Taylor bivariate polynomial of total degree m: (4.1) we have the bivariate Lagrange interpolation polynomial corresponding to the function f ∈ C(D) and to the nodes M k,r from the rectangle D = [a, b] × [c, d].

( 4 ( 1
.11) (T p f )(x, y) = c) i (y−d) j i!j! f (i,j) (c, d).If we assume that f belongs to the class C p+1 of functions having continuous all the partial derivatives of orders (p + 1 − i, i), (i = 0, 1, . . ., p + 1) in a neighborhood E c,d of the point (c, d), then the remainder R p f of the approximation formula of f by the bivariate Taylor polynomial (4.11) can be represented under the following form(R p f )(x, y) − u) n (x − c) ∂ ∂x + (y − d) ∂ ∂y (p+1) f c + (x − c)u, d + (y − d)u du,whenever the point (x, y) belongs to E c,d .This formula was deduced in the paper[8] of the first author.