Definition:Finite Difference Operator/Forward 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 forward difference operator on $f$ is defined as:
| \(\ds \map {\Delta^k f} {x_i}\) | \(=\) | \(\ds \map \Delta {\map {\Delta^{k - 1} f} {x_i} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \Delta^{k - 1} \map f {x_{i + 1} } - \Delta^{k - 1} \map f {x_i}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{s \mathop = 0}^k \paren {-1}^{k - s} \dbinom k s y_{i + s}\) |
for $i = 0, 1, 2, \ldots, n - k$
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Also see
- Results about the forward difference operator can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): finite differences
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): finite differences