TRIVARIATE APPROXIMATION OPERATORS ON CUBE BY PARAMETRIC EXTENSIONS by Teodora C ă tina
نویسنده
چکیده
The interpolation by polynomials or other functions is a rather old method in applied mathematics. In the paper of M. Gasca and T. Sauer [9] it is mentioned that the word ''interpolation'' has apparently been introduced by J.Wallis as early as 1655. Compared to this, polynomial interpolation in several variables is a relatively new topic and it probably only started in the second half of 19 century. In the same paper it is mentioned a statement of Andoyer regarding multivariate polynomial interpolation: ''It is clear that the interpolation of function of several variables does not demand any new principles because the fact that the variable was unique has not played frequently any role.” The time has contradicted this statement and multivariate polynomial interpolation has received constant attention and is today a basic subject in Approximation Theory and Numerical Analysis, with applications to many problems in Mathematics, Physics, Engineering, and so on. An approximation problem consists in expressing a group of values in terms of another group of values. There are two main ways to approximate a function of several variables: to extend the known results from the univariate case or to use specific approximation procedures for multivariate functions. Bivariate interpolation by the tensor product of univariate interpolation functions (that is, when the variables are treated separately) is the clasical approach to multivariate interpolation. This idea is impossible to use when the set of interpolation points is not a cartesian product grid. In this paper we consider the cartesian product grid case. It is known that the tensorial product of two univariate interpolation operators is a bivariate interpolation operator. So, if A and B are interpolation operators corresponding to the nodes s1,...,sn and respectively t1,...,tm then B ⊗ A is an interpolation operator corresponding to the grid of nodes m} , 1, j n; , 1, i | ) t , {(s j i ... = ... = .
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