Editing Finite difference (section)

==Generalizations==

*A '''generalized finite difference''' is usually defined as <math display="block">\Delta_h^\mu[f](x) = \sum_{k=0}^N \mu_k f(x+kh),</math> where {{math|1=''μ'' = (''μ''<sub>0</sub>, …, ''μ<sub>N</sub>'')}} is its coefficient vector. An '''infinite difference''' is a further generalization, where the finite sum above is replaced by an [[Series (mathematics)|infinite series]]. Another way of generalization is making coefficients {{math|''μ<sub>k</sub>''}} depend on point {{mvar|x}}: {{math|1=''μ<sub>k</sub>'' = ''μ<sub>k</sub>''(''x'')}}, thus considering '''weighted finite difference'''. Also one may make the step {{mvar|h}} depend on point {{mvar|x}}: {{math|1=''h'' = ''h''(''x'')}}. Such generalizations are useful for constructing different [[modulus of continuity]].
*The generalized difference can be seen as the polynomial rings {{math|''R''[''T<sub>h</sub>'']}}. It leads to difference algebras.
*Difference operator generalizes to [[Möbius inversion]] over a [[partially ordered set]].
*As a convolution operator: Via the formalism of [[incidence algebra]]s, difference operators and other Möbius inversion can be represented by [[convolution]] with a function on the poset, called the [[Möbius function (combinatorics)|Möbius function]] {{mvar|μ}}; for the difference operator, {{mvar|μ}} is the sequence {{nowrap|(1, −1, 0, 0, 0, …)}}.