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Pincherle derivative

## Summary

In mathematics, the Pincherle derivative[1] ${\displaystyle T'}$ of a linear operator ${\displaystyle T:\mathbb {K} [x]\to \mathbb {K} [x]}$ on the vector space of polynomials in the variable x over a field ${\displaystyle \mathbb {K} }$ is the commutator of ${\displaystyle T}$ with the multiplication by x in the algebra of endomorphisms ${\displaystyle \operatorname {End} (\mathbb {K} [x])}$. That is, ${\displaystyle T'}$ is another linear operator ${\displaystyle T':\mathbb {K} [x]\to \mathbb {K} [x]}$

${\displaystyle T':=[T,x]=Tx-xT=-\operatorname {ad} (x)T,\,}$

(for the origin of the ${\displaystyle \operatorname {ad} }$ notation, see the article on the adjoint representation) so that

${\displaystyle T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad \forall p(x)\in \mathbb {K} [x].}$

This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936).

## Properties

The Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators ${\displaystyle S}$  and ${\displaystyle T}$  belonging to ${\displaystyle \operatorname {End} \left(\mathbb {K} [x]\right),}$

1. ${\displaystyle (T+S)^{\prime }=T^{\prime }+S^{\prime }}$ ;
2. ${\displaystyle (TS)^{\prime }=T^{\prime }\!S+TS^{\prime }}$  where ${\displaystyle TS=T\circ S}$  is the composition of operators.

One also has ${\displaystyle [T,S]^{\prime }=[T^{\prime },S]+[T,S^{\prime }]}$  where ${\displaystyle [T,S]=TS-ST}$  is the usual Lie bracket, which follows from the Jacobi identity.

The usual derivative, D = d/dx, is an operator on polynomials. By straightforward computation, its Pincherle derivative is

${\displaystyle D'=\left({d \over {dx}}\right)'=\operatorname {Id} _{\mathbb {K} [x]}=1.}$

This formula generalizes to

${\displaystyle (D^{n})'=\left({{d^{n}} \over {dx^{n}}}\right)'=nD^{n-1},}$

by induction. This proves that the Pincherle derivative of a differential operator

${\displaystyle \partial =\sum a_{n}{{d^{n}} \over {dx^{n}}}=\sum a_{n}D^{n}}$

is also a differential operator, so that the Pincherle derivative is a derivation of ${\displaystyle \operatorname {Diff} (\mathbb {K} [x])}$ .

When ${\displaystyle \mathbb {K} }$  has characteristic zero, the shift operator

${\displaystyle S_{h}(f)(x)=f(x+h)\,}$

can be written as

${\displaystyle S_{h}=\sum _{n\geq 0}{{h^{n}} \over {n!}}D^{n}}$

by the Taylor formula. Its Pincherle derivative is then

${\displaystyle S_{h}'=\sum _{n\geq 1}{{h^{n}} \over {(n-1)!}}D^{n-1}=h\cdot S_{h}.}$

In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars ${\displaystyle \mathbb {K} }$ .

If T is shift-equivariant, that is, if T commutes with Sh or ${\displaystyle [T,S_{h}]=0}$ , then we also have ${\displaystyle [T',S_{h}]=0}$ , so that ${\displaystyle T'}$  is also shift-equivariant and for the same shift ${\displaystyle h}$ .

The "discrete-time delta operator"

${\displaystyle (\delta f)(x)={{f(x+h)-f(x)} \over h}}$

is the operator

${\displaystyle \delta ={1 \over h}(S_{h}-1),}$

whose Pincherle derivative is the shift operator ${\displaystyle \delta '=S_{h}}$ .