Type of derivative of a linear operator
In mathematics, the Pincherle derivative[1]
of a linear operator
on the vector space of polynomials in the variable x over a field
is the commutator of
with the multiplication by x in the algebra of endomorphisms
. That is,
is another linear operator
![{\displaystyle T':=[T,x]=Tx-xT=-\operatorname {ad} (x)T,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adeaa560041ded4e7cc9ef6d1ee77550949a01d8)
(for the origin of the
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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25baff00a3c5ca6de0647d4daf76c6622b41b421)
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
and
belonging to
;
where
is the composition of operators.
One also has
where
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/067676cf06f192b93b47d2122716b3e8609b6f25)
This formula generalizes to
![{\displaystyle (D^{n})'=\left({{d^{n}} \over {dx^{n}}}\right)'=nD^{n-1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d633fcef0512ff60e3b9e58fe1d77e96eb85dfcb)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c46364861aa39a3cb6762285a6ab2c0eab7c41f0)
is also a differential operator, so that the Pincherle derivative is a derivation of
.
When
has characteristic zero, the shift operator
![{\displaystyle S_{h}(f)(x)=f(x+h)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/917a0a9cb0b633fc76bed268fffbb9c88a01668f)
can be written as
![{\displaystyle S_{h}=\sum _{n\geq 0}{{h^{n}} \over {n!}}D^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98bd41c927a1e86e3da2920907915b10de67a70d)
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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d75f4d1fdef138b04f73d552c3c4aefee8207fa6)
In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars
.
If T is shift-equivariant, that is, if T commutes with Sh or
, then we also have
, so that
is also shift-equivariant and for the same shift
.
The "discrete-time delta operator"
![{\displaystyle (\delta f)(x)={{f(x+h)-f(x)} \over h}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d45ef8f986a6410e3ba0cdd986a4778b6d099082)
is the operator
![{\displaystyle \delta ={1 \over h}(S_{h}-1),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df5fed7433c9154c1e23db4e816b6ffe5d678db9)
whose Pincherle derivative is the shift operator
.
See also
References
External links