Definition:Finite Difference Operator/Backward Difference/kth
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Definition
Let $f: \R \to \R$ be a real function.
Let $y = \map f x$ have known values:
- $y_k = \map f {x_k}$
for $x_k \in \set {x_0, x_1, \ldots, x_n}$ defined as:
- $x_k = x_0 + k h$
for some $h \in \R_{>0}$.
The $k$th backward difference operator on $f$ is defined as:
| \(\ds \map {\nabla^k f} {x_i}\) | \(=\) | \(\ds \map \nabla {\map {\nabla^{k - 1} f} {x_i} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \nabla^{k - 1} \map f {x_i} - \nabla^{k - 1} \map f {x_{i - 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{s \mathop = 0}^k \paren {-1}^{k - s} \dbinom k s y_{i - s}\) |
for $i = k, k + 1, k + 2, \ldots, n$
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Also see
- Results about the backward difference operator can be found here.