Pincherle Derivative

In mathematics, the Pincherle derivative $T'$ of a linear operator T: \mathbb \to \mathbb /math> on the vector space of polynomials in the variable ''x'' over a field $\mathbb$ is the commutator of $T$ with the multiplication by ''x'' in the algebra of endomorphisms \operatorname(\mathbb . That is, $T'$ is another linear operator T': \mathbb \to \mathbb /math>

:T' := ,x= Tx-xT = -\operatorname(x)T,\,

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

:T'\=T\-xT\\qquad\forall p(x)\in \mathbb

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 $S$ and $T$ belonging to \operatorname\left( \mathbb \right),

#$(T + S)^\prime = T^\prime + S^\prime$;
#$(TS)^\prime = T^\prime\!S + TS^\prime$ where $TS = T \circ S$ is the composition of operators.

One also has ,S = ^, S+ , S^/math> where ,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

: $D'= \left(\right)' = \operatorname_ = 1.$

This formula generalizes to

: $(D^n)'= \left(\right)' = nD^,$

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

: $\partial = \sum a_n = \sum a_n D^n$

is also a differential operator, so that the Pincherle derivative is a derivation of \operatorname(\mathbb K .

When $\mathbb$ has characteristic zero, the shift operator

: $S_h(f)(x) = f(x+h) \,$

can be written as

: $S_h = \sum_ D^n$

by the Taylor formula. Its Pincherle derivative is then

: $S_h' = \sum_ D^ = h \cdot S_h.$

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

If ''T'' is shift-equivariant, that is, if ''T'' commutes with ''S''_''h'' or ,S_h= 0, then we also have ',S_h= 0, so that $T'$ is also shift-equivariant and for the same shift $h$.

The "discrete-time delta operator"

: $(\delta f)(x) =$

is the operator

: $\delta = (S_h - 1),$

whose Pincherle derivative is the shift operator $\delta' = S_h$.

See also

*Commutator
* Delta operator
* Umbral calculus

References

{{Reflist

External links

*Weisstein, Eric W. "
Pincherle Derivative
'". From MathWorld—A Wolfram Web Resource.
*

' at the MacTutor History of Mathematics archive.

Differential algebra