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Delta operator

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated November 12, 2021

In mathematics, a delta operator is a shift-equivariant linear operator Q\colon \mathbb {K} [x]\longrightarrow \mathbb {K} [x] on the vector space of polynomials in a variable x over a field \mathbb {K} that reduces degrees by one.

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

{(Qg)(x)=(Qf)(x+a)}.\,

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

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

Sometimes a delta operator is defined to be a shift-equivariant linear transformation on polynomials in x that maps 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 \mathbb {K} has characteristic zero, since shift-equivariance is a fairly strong condition.

01Examples

(\Delta f)(x)=f(x+1)-f(x)\,
is a delta operator.
  • Differentiation with respect to x, written as D, is also a delta operator.
  • Any operator of the form
\sum _{k=1}^{\infty }c_{k}D^{k}
(where Dn(ƒ) = ƒ(n) is the nth derivative) with 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
\Delta =e^{D}-1=\sum _{k=1}^{\infty }{\frac {D^{k}}{k!}}.
{(\delta f)(x)={{f(x+\Delta t)-f(x)} \over {\Delta t}}},
the Euler approximation of the usual derivative with a discrete sample time \Delta t. The delta-formulation obtains a significant number of numerical advantages compared to the shift-operator at fast sampling.

02Basic polynomials

Every delta operator Q has a unique sequence of "basic polynomials", a polynomial sequence defined by three conditions:

  • p_{0}(x)=1;
  • p_{n}(0)=0;
  • (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, 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.

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Sources and credits

This article is adapted from the Wikipedia article “Delta operator”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Pincherle derivative

In mathematics, the Pincherle derivative T ′ {\displaystyle T'} of a linear operator T : K [ x ] → K [ x ] {\displaystyle T:\mathbb {K} [x]\to \mathbb {K} [x]} on the vector space of polynomials in the variable x over a field K {\displaystyle \mathbb {K} } is the commutator of T {\displaystyle T} with the multiplication by x in the algebra of endomorphisms End ⁡ {\displaystyle \operatorname {End} (\mathbb {K} [x])} . That is, T ′ {\displaystyle T'} is another linear operator T ′ : K [ x ] → K [ x ] {\displaystyle T':\mathbb {K} [x]\to \mathbb {K} [x]} T ′ := [ T , x ] = T x − x T = − ad ⁡ ( x ) T , {\displaystyle T':=[T,x]=Tx-xT=-\operatorname {ad} (x)T,\,} (for the origin of the ad {\displaystyle \operatorname {ad} } notation, see the article on the adjoint representation) so that T ′ { p ( x ) } = T { x p ( x ) } − x T { p ( x ) } ∀ p ( x ) ∈ K [ x ] .

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Umbral calculus

The term umbral calculus has two related but distinct meanings. In mathematics, before the 1970s, umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to prove them.