A new non-polynomial solution to multivariate Hermite-Birkhoff interpolation

A new solution to the multivariate Hermite-Birkhoff interpolation problem is presented. The classical approach to this problem consists in constructing the minimum degree polynomial, which coincides with the prescribed function and derivative values at the sample points. Here the interpolant is represented as a truncated Multipoint Taylor (MT) series. A MT series can be regarded as an extension to multiple points of the ordinary (one point) Taylor series. The constructed MT series converges when (i) the number of derivatives tends to infinity, while the number of sample points remains finite, or (ii) when the spacing between sample points tends to zero, while the number of derivatives remains finite. The MT series region of convergence depends on the support of its basis functions and the location of the singularities of the function represented. Examples are given of the construction of interpolants in one and two dimensions and for different samples grid types. A comparison between the numerical results achieved with this new method and those obtained with onedimensional polynomial Hermite-Birkhoff interpolation is made. Our interpolant does not suffer from the ill conditioning seen with the classical polynomial solution for certain data configurations. It is always numerically stable in its region of convergence. In addition, it is computationally much more efficient than the polynomial approach, because no basis function construction phase is required. Finally, the presented construction is applicable to interpolation in any number of dimensions and could also be regarded as a finite element data representation, which uses both functional values as well as derivative values.