Divided differences for functions of two variables for irregularly spaced arguments

Abstract

Divided differences forf (x, y) for completely irregular spacing of points (x _i ,y _i ) are developed here by a natural generalization of Newton's scheme. Existing bivariate schemes either iterate the one-dimensional scheme, thus constraining (x _i ,y _i ) to be at corners of rectangles, or give polynomials Σa _jk x ^j y ^k having more coefficients than interpolation conditions. Here the generalizedn ^th divided difference is defined by (1)$\left[ {01... n} \right] = \sum\limits_{i = 0}^n {A_i f\left( {x_i , y_i } \right)}$ where (2)$\sum\limits_{i = 0}^n {A_i x_i^j , y_i^k = 0}$, and 1 for the last or (n+1)^th equation, for every (j, k) wherej+k=0, 1, 2,... in the usual ascending order. The gen. div. diff. [01...n] is symmetric in (x _i ,y _i ), unchanged under translation, 0 forf (x, y) an, ascending binary polynomial as far asn terms, degree-lowering with respect to (X, Y) whenf(x, y) is any polynomialP(X+x, Y+y), and satisfies the 3-term recurrence relation (3) [01...n]=λ{[1...n]−[0...n−1]}, where (4) λ= |1...n|·|01...n−1|/|01...n|·|1...n−1|, the |...i...| denoting determinants inx ^j _i y ^k _i . The generalization of Newton's div. diff. formula is (5)

$$\begin{gathered} f\left( {x, y} \right) = f\left( {x_0 , y_0 } \right) - \frac{{\left| {\alpha 0} \right|}}{{\left| 0 \right|}}\left[ {01} \right] + \frac{{\left| {\alpha 01} \right|}}{{\left| {01} \right|}}\left[ {012} \right] - \frac{{\left| {\alpha 012} \right|}}{{\left| {012} \right|}}\left[ {0123} \right] + \cdots + \hfill \\ + \left( { - 1} \right)^n \frac{{\left| {\alpha 01 \ldots n - 1} \right|}}{{\left| {01 \ldots n - 1} \right|}}\left[ {01 \ldots n} \right] + \left( { - 1} \right)^{n + 1} \frac{{\left| {\alpha 01 \ldots n} \right|}}{{\left| {01 \ldots n} \right|}}\left[ {01 \ldots n} \right], \hfill \\ \end{gathered} $$

the α denotingx ⁱ y ^k terms. The right member of (5) without the last or remainder term, =f(x _i ,y _i ) for (x, y)=(x _i ,y _i ),i=0, 1, ...,n. Gen. div. diffs. having most of the preceding properties may be defined for any number of dimensions, and other than polynomial approximation, by replacingx ^j y ^k in (2) by other functions. confluent forms of (1) and (5) which usually exist (sometimes not), and generally (not always) depend upon the direction of confluence, are investigated up ton=5, for complete confluence at one point and confluence at two distinct points, those found giving (in latter case, also by an earlier theorem) bivariate osculatory interpolation formulas.