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:
 * $$\Delta f \left({x}\right) \ \stackrel {\mathbf {def}} {=\!=} \ f \left({x + 1}\right) - f \left({x}\right)$$

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

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

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

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

Also see
Compare with derivative.