# Delta operator

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Delta_operator
> Markdown URL: https://mediated.wiki/source/Delta_operator.md
> Source: https://en.wikipedia.org/wiki/Delta_operator
> Source revision: 1054932186
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [mathematics](/source/Mathematics), a **delta operator** is a shift-equivariant [linear operator](/source/Linear_operator) Q : K [ x ] ⟶ K [ x ] {\displaystyle Q\colon \mathbb {K} [x]\longrightarrow \mathbb {K} [x]} on the [vector space](/source/Vector_space) of [polynomials](/source/Polynomial) in a variable x {\displaystyle x} over a [field](/source/Field_(mathematics)) K {\displaystyle \mathbb {K} } that reduces [degrees](/source/Degree_of_a_polynomial) by one.

To say that Q {\displaystyle Q} is **shift-equivariant** means that if g ( x ) = f ( x + a ) {\displaystyle g(x)=f(x+a)} , then

- ( Q g ) ( x ) = ( Q f ) ( x + a ) . {\displaystyle {(Qg)(x)=(Qf)(x+a)}.\,}

In other words, if f {\displaystyle f} is a "shift" of g {\displaystyle g} , then Q f {\displaystyle Qf} is also a shift of Q g {\displaystyle Qg} , and has the same "shifting vector" a {\displaystyle a} .

To say that an operator *reduces degree by one* means that if f {\displaystyle f} is a polynomial of degree n {\displaystyle n} , then Q f {\displaystyle Qf} is either a polynomial of degree n − 1 {\displaystyle n-1} , or, in case n = 0 {\displaystyle n=0} , Q f {\displaystyle Qf} is 0.

Sometimes a *delta operator* is defined to be a shift-equivariant linear transformation on polynomials in x {\displaystyle x} that maps x {\displaystyle x} 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 K {\displaystyle \mathbb {K} } has [characteristic](/source/Characteristic_(algebra)) zero, since shift-equivariance is a fairly strong condition.

## Examples

- The forward [difference operator](/source/Difference_operator)

- - ( Δ f ) ( x ) = f ( x + 1 ) − f ( x ) {\displaystyle (\Delta f)(x)=f(x+1)-f(x)\,}

- is a delta operator.

- [Differentiation](/source/Derivative) with respect to *x*, written as *D*, is also a delta operator.

- Any operator of the form

- - ∑ k = 1 ∞ c k D k {\displaystyle \sum _{k=1}^{\infty }c_{k}D^{k}}

- (where *D**n*(ƒ) = ƒ(*n*) is the *n*th [derivative](/source/Derivative)) with c 1 ≠ 0 {\displaystyle c_{1}\neq 0} 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 - Δ = e D − 1 = ∑ k = 1 ∞ D k k ! . {\displaystyle \Delta =e^{D}-1=\sum _{k=1}^{\infty }{\frac {D^{k}}{k!}}.}

- The generalized derivative of [time scale calculus](/source/Time_scale_calculus) which unifies the forward difference operator with the derivative of [standard calculus](/source/Calculus) is a delta operator.

- In [computer science](/source/Computer_science) and [cybernetics](/source/Cybernetics), the term "discrete-time delta operator" (δ) is generally taken to mean a difference operator

- - ( δ f ) ( x ) = f ( x + Δ t ) − f ( x ) Δ t , {\displaystyle {(\delta f)(x)={{f(x+\Delta t)-f(x)} \over {\Delta t}}},}

- the [Euler approximation](/source/Euler_approximation) of the usual derivative with a discrete sample time Δ t {\displaystyle \Delta t} . The delta-formulation obtains a significant number of numerical advantages compared to the shift-operator at fast sampling.

## Basic polynomials

Every delta operator *Q {\displaystyle Q}* has a unique sequence of "basic polynomials", a [polynomial sequence](/source/Polynomial_sequence) defined by three conditions:

- p 0 ( x ) = 1 ; {\displaystyle p_{0}(x)=1;}

- p n ( 0 ) = 0 ; {\displaystyle p_{n}(0)=0;}

- ( Q p n ) ( x ) = n p n − 1 ( x ) for all n ∈ N . {\displaystyle (Qp_{n})(x)=np_{n-1}(x){\text{ for all }}n\in \mathbb {N} .}

Such a sequence of basic polynomials is always of [binomial type](/source/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](/source/Sheffer_sequence)—a more general concept.

## See also

- [Pincherle derivative](/source/Pincherle_derivative)

- [Shift operator](/source/Shift_operator)

- [Umbral calculus](/source/Umbral_calculus)

## References

- Nikol'Skii, Nikolai Kapitonovich (1986), *Treatise on the shift operator: spectral function theory*, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [ISBN](/source/ISBN_(identifier)) [978-0-387-15021-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-15021-5)

## External links

- [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Delta Operator"](https://mathworld.wolfram.com/DeltaOperator.html). *[MathWorld](/source/MathWorld)*.

---
Adapted from the Wikipedia article [Delta operator](https://en.wikipedia.org/wiki/Delta_operator) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Delta_operator?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.