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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.

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I'm not sure why you've got the stochastic calculus and numerical analysis tags as you're talking about finite differences. The Wikipedia page on the topic provides you with answers to your single-variable question and references for the multi-variable question. en.wikipedia.org/wiki/Finite_difference –  Ryan Budney Nov 14 '10 at 17:27
I've reordered the tags and changed stochastic-calculus to stochastic-processes. Why stochastic processes, because the paper and the idea of generalizing finite differences arises in stochastic processes; and finite differences are commonly associated with the field of numerical-analysis. My question is not about finite differences alone (I have read the Wikipedia article) but about the linear combination of weights of finite differences and the conditions under which the linear combination of weights is zero. –  Olumide Nov 14 '10 at 21:49
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