Global and Radial Interpolation

For many applications, interpolations of functions of two or three variables defined in two-and three-dimensional domains must be considered. For example, global interpolations in two- and three-dimensional systems are analogous to polynomial interpolation in onedimensional systems however, global interpolants do not exist in 2- and 3D. This is a big drawback in numerical analysis because a basic tool available for one variable is not available for multivariable approximation [21], The best developed aspect of this theory is that of piecewise polynomial approximation, associated with finite element and finite volume approximations for partial differential equations, which will be examined in detail in Chapters 9 and 10. 

The matrix = [ (xj)] is called the Gram matrix, when the matrix is non-singular (has an inverse). If the matrix is non-singular, the equation I a = f has a unique solution however, according to Golberg [21], this fails to be true even in very simple cases [21]. 

Linear interpolant for three points in a two dimensional domain. We want to obtain a linear interpolation for a function f(x, y) at three points (Xj,yj) for j = 1,2, 3. The proposed interpolant will be, 

In order for this matrix to be singular, the three points must be non-collinear. Otherwise an infinite number of planes exist that pass through the given line. It is difficult 

Global basis functions. Common global basis functions, where the interpolation functions for multi-dimensional domains can be obtained, come from expansions of Pascal s triangle. In 2D, Pascal s triangle is defined by, 

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