Pincherle derivative

In mathematics, the Pincherle derivative[1] T’ of a linear operator T:K[x] → K[x] on the vector space of polynomials in the variable x over a field K is the commutator of T with the multiplication by x in the algebra of endomorphisms End(K[x]). That is, T’ is another linear operator T’:K[x] → K[x]

(for the origin of the ad notation, see the article on the adjoint representation) so that

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

PropertiesEdit

The Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators   and   belonging to  

  1.   ;
  2.   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

 

This formula generalizes to

 

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

 

is also a differential operator, so that the Pincherle derivative is a derivation of  .

When   has characteristic zero, the shift operator

 

can be written as

 

by the Taylor formula. Its Pincherle derivative is then

 

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"

 

is the operator

 

whose Pincherle derivative is the shift operator  .

See alsoEdit

ReferencesEdit

  1. ^ Rota, Gian-Carlo; Mullin, Ronald (1970). Graph Theory and Its Applications. Academic Press. pp. 192. ISBN 0123268508.

External linksEdit