In mathematics, the Pincherle derivative[1]
T'
T:K[x]\toK[x]
K
T
\operatorname{End}(K[x])
T'
T':K[x]\toK[x]
T':=[T,x]=Tx-xT=-\operatorname{ad}(x)T,
(for the origin of the
\operatorname{ad}
T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\} \forallp(x)\inK[x].
This concept is named after the Italian mathematician Salvatore Pincherle (1853 - 1936).
The Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators
S
T
\operatorname{End}\left(K[x]\right),
(T+S)\prime=T\prime+S\prime
(TS)\prime=T\primeS+TS\prime
TS=T\circS
One also has
[T,S]\prime=[T\prime,S]+[T,S\prime]
[T,S]=TS-ST
The usual derivative, D = d/dx, is an operator on polynomials. By straightforward computation, its Pincherle derivative is
D'=\left({d\over{dx}}\right)'=\operatorname{Id}K[x]=1.
This formula generalizes to
(Dn)'=\left({{dn}\over{dxn}}\right)'=nDn-1,
by induction. This proves that the Pincherle derivative of a differential operator
\partial=\suman{{dn}\over{dxn}}=\sumanDn
is also a differential operator, so that the Pincherle derivative is a derivation of
\operatorname{Diff}(K[x])
When
K
Sh(f)(x)=f(x+h)
can be written as
Sh=\sumn{{hn}\over{n!}}Dn
by the Taylor formula. Its Pincherle derivative is then
Sh'=\sumn{{hn}\over{(n-1)!}}Dn-1=h ⋅ Sh.
In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars
K
If T is shift-equivariant, that is, if T commutes with Sh or
[T,Sh]=0
[T',Sh]=0
T'
h
The "discrete-time delta operator"
(\deltaf)(x)={{f(x+h)-f(x)}\overh}
is the operator
\delta={1\overh}(Sh-1),
whose Pincherle derivative is the shift operator
\delta'=Sh