Linear combination of finite difference weights and generalized increments

2010-11-14T15:41:49Z

According to a paper I'm reading ("Linear estimation of non stationary spatial phenomena", by P. Delfiner) the weights $\lambda_i$ and $\lambda_{i - 1}$ of a first order finite difference $Z(x_i) - Z(x_{i - 1})$ satisfy the relation

$\lambda_i + \lambda_{i - 1} = 1 + (-1) = 0$

and the weights of second order finite differences $Z(x_{i + 1}) + 2 Z(x_i) - Z(x_{i - 1})$ satisfy

$\lambda_{i + 1} + \lambda_i + \lambda_{i - 1} = 1 + (-2) + 1 = 0$

$\lambda_{i + 1} x_{i + 1} + \lambda_i x_{i} + \lambda_{i - 1} x_{i - 1} = (1)(1) + (2)(0) + (1)(-1) = 0$

assuming $x_{i + 1} = 1$, $x_{i} = 0$ and $x_{i + 1} = -1$. In general the weights of nth order finite differences satisfy

$\sum_{i = 0}^m \lambda_i x_i^r = 0$

for all $0 \le r < n - 1$, where $m$ is the number of points. Furthermore, in two dimensions, the weights of weights of nth order finite differences satisfy the linear combination

$\sum_{i = 0}^m \lambda_i x_{i1}^r x_{i2}^s = 0$

for all $r , s \ge 0$ and $r + s \: \le \: n - 1$, and in three dimensions, the weights of nth order finite differences satisfy the linear combination

$\sum_{i = 0}^m \lambda_i x_{i1}^r x_{i2}^s x_{i2}^t = 0$

for all $r , s , t \ge 0$ and $r + s + t \: \le \: n - 1$

My question is under what conditions do these relations hold? For example, must the $Z(.)$'s be equally spaced? Also, I'm a bit worried that as no proof is given, the conditions appear to me to be mere observations. I've not found any information about these conditions in literature.

Postscript: If it matters, the article uses these set of conditions in order to assert that

$\sum_{i = 0}^m \lambda_i Z(x_i)$

is a generalization of the finite differencing operation (it prefers the term generalized increment) of order n if all linear combinations of the $\lambda$'s up to n sum to zero as described above.

Linear combination of finite difference weights and generalized increments - MathOverflow

Linear combination of finite difference weights and generalized increments