A generalization of Hermite's interpolation formula in two variables

Abstract

Spitzbart [1] has considered a generalization of Hermite's interpolation formula in one variable and has obtained a polynomial p(x) of degree n + Σnj=0 = rj in x which interpolates to the values of a function and its derivatives up to order rj at xj, j = 0, 1,···n. Ahlin [2] has considered a bivariate generalization of Hermite's interpolation formula. He has developed a bivariate osculatory interpolation polynomial which agrees with f(x, y) and its partial and mixed partial derivatives up to a specified order at each of the nodes of a Cartesian grid. However, the above interpolation problem considered by Ahlin assumes that the values of partial and mixed partial derivatives of the same fixed order k – 1 are available at every point of the rectangular grid. It may also be observed that Ahlin's formula is essentially a Cartesian product of a special case of Spitzbart's formula in one variable.

Simultaneous Approximation of Measurement Values and Derivative Data using Discrete Orthogonal Polynomials

“…A generalized version for arbitrary orders of derivatives at given points can be found in [3]. This idea is extended to bivariate functions in [4]. An extensive study on multivariate Hermite interpolation can be found in [5].…”

On the confluent Vandermonde matrix calculation algorithm

“…However, in Ahlin [2] there is an additional restriction that at each node of a 2D grid we consider the same total order of derivatives or partial derivatives. This restriction removes the article Chawla, Jayarajan [11] from 1974.…”

Classical multivariate Hermite coordinate interpolation in n-dimensional grid

“…The 1D Hermite interpolant of Theorem 1.1 can be easily extended to multivariate interpolation. Namely, in [7] the two dimensional (2D) interpolation is considered. In [8] we extended our previous work [6] into two dimensions (2D), deriving closed-form expression that was again much more compact than [9,7] and applicable in the case of support points arranged on a non-equidistant grid.…”

“…Namely, in [7] the two dimensional (2D) interpolation is considered. In [8] we extended our previous work [6] into two dimensions (2D), deriving closed-form expression that was again much more compact than [9,7] and applicable in the case of support points arranged on a non-equidistant grid. For both the 1D and 2D cases, we provided means for computationally efficient implementations of the proposed Hermite interpolating polynomials that achieved computational complexity comparable to other popular and much simpler interpolation techniques, such as cubic splines, whereas the measured error when applied to clinical medical images was superior.…”