Editing Delta operator

In [[mathematics]], a '''delta operator''' is a shift-equivariant [[linear operator]] <math>Q\colon\mathbb{K}[x] \longrightarrow \mathbb{K}[x]</math> on the [[vector space]] of [[polynomial]]s in a variable <math>x</math> over a [[field (mathematics)|field]] <math>\mathbb{K}</math> that reduces [[degree of a polynomial|degrees]] by one.

To say that <math>Q</math> is '''shift-equivariant''' means that if <math>g(x) = f(x + a)</math>, then

:<math>{ (Qg)(x) = (Qf)(x + a)}.\,</math>

In other words, if <math>f</math> is a "shift" of <math>g</math>, then <math>Qf</math> is also a shift of <math>Qg</math>, and has the same "shifting vector" <math>a</math>.

To say that an operator ''reduces degree by one'' means that if <math>f</math> is a polynomial of degree <math>n</math>, then <math>Qf</math> is either a polynomial of degree <math>n-1</math>, or, in case <math>n = 0</math>, <math>Qf</math> is 0.

Sometimes a ''delta operator'' is defined to be a shift-equivariant linear transformation on polynomials in <math>x</math> that maps <math>x</math> to a nonzero constant.  Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition when <math>\mathbb{K}</math> has [[characteristic (algebra)|characteristic]] zero, since shift-equivariance is a fairly strong condition.

==Examples==

* The forward [[difference operator]]

:: <math> (\Delta f)(x) = f(x + 1) - f(x)\, </math>

:is a delta operator.

* [[Derivative|Differentiation]] with respect to ''x'', written as ''D'', is also a delta operator.
* Any operator of the form
::<math>\sum_{k=1}^\infty c_k D^k</math>
: (where ''D''<sup>''n''</sup>(&fnof;) = &fnof;<sup>(''n'')</sup> is the ''n''<sup>th</sup> [[derivative]]) with <math>c_1\neq0</math> is a delta operator.  It can be shown that all delta operators can be written in this form.  For example, the difference operator given above can be expanded as
::<math>\Delta=e^D-1=\sum_{k=1}^\infty \frac{D^k}{k!}.</math>

* The generalized derivative of [[time scale calculus]] which unifies the forward difference operator with the derivative of [[calculus|standard calculus]] is a delta operator.
* In [[computer science]] and [[cybernetics]], the term "discrete-time delta operator" (&delta;) is generally taken to mean a difference operator

:: <math>{(\delta f)(x) = {{ f(x+\Delta t) - f(x) }  \over {\Delta t} }}, </math>

: the [[Euler approximation]] of the usual derivative with a discrete sample time <math>\Delta t</math>. The delta-formulation obtains a significant number of numerical advantages compared to the shift-operator at fast sampling.

==Basic polynomials==

Every delta operator ''<math>Q</math>'' has a unique sequence of "basic polynomials", a [[polynomial sequence]] defined by three conditions:

* <math>p_0(x)=1 ;</math>
* <math>p_{n}(0)=0;</math>
* <math>(Qp_n)(x)=np_{n-1}(x) \text{ for all } n \in \mathbb N.</math>

Such a sequence of basic polynomials is always of [[binomial type]], and it can be shown that no other sequences of binomial type exist.  If the first two conditions above are dropped, then the third condition says this polynomial sequence is a [[Sheffer sequence]]—a more general concept.

== See also ==

* [[Pincherle derivative]]
* [[Shift operator]]
* [[Umbral calculus]]

== References ==
* {{Citation | last1=Nikol'Skii | first1=Nikolai Kapitonovich | title=Treatise on the shift operator: spectral function theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-15021-5 | year=1986}}

== External links ==
* {{MathWorld|title=Delta Operator|urlname=DeltaOperator}}

[[Category:Linear algebra]]
[[Category:Polynomials]]
[[Category:Finite differences]]