Finite Difference - Generalizations

Generalizations

• A generalized finite difference is usually defined as

where is its coefficients vector. An infinite difference is a further generalization, where the finite sum above is replaced by an infinite series. Another way of generalization is making coefficients depend on point :, thus considering weighted finite difference. Also one may make step depend on point : . Such generalizations are useful for constructing different modulus of continuity.

• Difference operator generalizes to Möbius inversion over a partially ordered set.

• As a convolution operator: Via the formalism of incidence algebras, difference operators and other Möbius inversion can be represented by convolution with a function on the poset, called the Möbius function μ; for the difference operator, μ is the sequence (1, −1, 0, 0, 0, ...).