Definition:Finite Difference Operator

Definition
Let $f: \R \to \R$ be a real function.

The (finite) difference operator on $f$ comes in a number of forms, as follows.

Forward Difference
The forward difference operator is defined as:
 * $\Delta f \left({x}\right) := f \left({x + 1}\right) - f \left({x}\right)$

Backward Difference
The backward difference operator is defined as:
 * $\nabla f \left({x}\right) := f \left({x}\right) - f \left({x - 1}\right)$

Generalized Forward Difference
The forward difference operator is defined as:
 * $\Delta_h f \left({x}\right) := f \left({x + h}\right) - f \left({x}\right)$

Generalized Backward Difference
The backward difference operator is defined as:
 * $\nabla_h f \left({x}\right) := f \left({x}\right) - f \left({x - h}\right)$

Central Difference
The central difference operator is defined as:
 * $\delta_h f \left({x}\right) := f \left({x + \dfrac h 2}\right) - f \left({x - \dfrac h 2}\right)$

Also see
Compare with derivative.