HOMOGENEOUS NUMERICAL CUBATURE FORMULAS OF INTERPOLATORY

. In this paper we construct homogeneous numerical cubature formulas based on some numerical multivariate interpolation schemes. MSC 2000. 65D32.


INTRODUCTION
Let D be a given domain in R 2 , f : D → R an integrable function on D and Λ := {λ 1 f, . . ., λ N f } some given information on f .Next, one suppose that λ i f are values of f or of certain of its derivatives at some points of D, called nodes.
One considers the cubature formula where A i , i = 1, . . ., N are its coefficients and R N (f ) is the remainder term.
The coming problem is to find the parameters of such a cubature formula (coefficients, nodes) and to study the remainder term.
The most results has been obtained when D is a regular domain in R 2 (rectangle, triangle) and the information (data) are regularly spaced.At this class of cubature procedure belong the tensorial product and the cubature sum rules. Let . ., m} and Λ y := {λ y j f | j = 0, 1, . . ., n}, m, n ∈ N are given sets of information on f with regard to x respectively y, one considers the quadrature formulas and where the quadrature rules Q x 1 and Q y 1 are given by It is easy to check the following decomposition of the double integral operator I xy (1) ) and The identities (1) and (2) generate so called product cubature formula (3) f, respectively the boolean-sum cubature formula (4) 3) and (4) it follows that the approximation order of the product formula is min{p 1 , q 1 } while the approximation order of the boolean-sum formula is Hence, the boolean-sum cubature rules has the remarkable property regarding its highest approximation order.
Otherwise, the boolean-sum formula contains the simple integrals I x f , respectively I y f .But, this simple integrals can be approximated, in a second level of approximation, using new quadrature procedures.
From (4), one obtains ( 5) and Q y 2 are the quadrature rules used in the second level of approximation and R x 2 , R y 2 are the corresponding remainder operators.As can be seen The quadrature rules Q x 2 and Q y 2 can be chosen in many ways.First of all, it depends on the given information of the function f .
A natural way to choose them is such that the approximation order of the initial boolean-sum formula to be preserved.It is obvious that its approximation order cannot be increased.Definition 1.A cubature formula of the form (5) derived from the booleansum formula (4) which preserves its approximation order is called a consistent cubature formula.
Remark 1.The cubature formula ( 5) is consistent if the orders p 2 and q 2 of the quadrature procedures Q x 2 , respectively Q y 2 , used in the second level of approximation, satisfy the inequalities As the approximation order of the boolean-sum cubature cannot be increased, it is preferable to choose the quadrature procedures Q x 2 and Q y 2 such that each term of the remainder from (5) to have the same order of approximation.
Definition 2. A cubature formula, of the form (5), of which each term of the remainder has the same order of approximation is called a homogeneous cubature formula.
Remark 2. The cubature formula ( 5) is homogeneous if For example, let be and the gaussian quadrature rules.Then for boolean-sum cubature formula, we have where R S (f ) = h 6 576 f (2,2) (ξ, η).In order to get a homogeneous numerical cubature formula we must use, in a second level of approximation, some quadrature rules then we have the homogeneous cubature formula Next, we try to construct such homogeneous cubature formulas using interpolation formulas derived from a boolean-sum interpolation formula and not only.
It is know [2] that if P x 1 and P y 1 are univariate interpolation operators, from the corresponding boolean-sum formula f can be derived using in a second level of approximation, some new operators P x 2 and P y 2 , a numerical approximation formula, i.e. ( 8) where S x 1 f and S x 3 f the linear respectively the cubic spline that interpolates f with regard to Λ x and S y 1 f, S y 3 f the corresponding splines that interpolates f with regard to Λ y , i.e.
where s 1 i , s 3 i , s 1 j , s 3 j are the corresponding cardinal splines.
Theorem 2. Let f be an integrable function on D h . where Proof.The bivariate interpolation formula (8 Remark 3. Taking into account that if is an optimal quadrature formula in sense of Sard [7], formula ( 9) is also an almost optimal cubature formula [9].Now it is easy to verify that S x 1 e i = e i for i = 0, 1 and S x 1 e 2 = e 2 , with e i (x) = x i .