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In mathematics, the Pincherle derivative 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
(for the origin of the notation, see the article on the adjoint representation) so that
This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936).
- where is the composition of operators.
The usual derivative, D = d/dx, is an operator on polynomials. By straightforward computation, its Pincherle derivative is
This formula generalizes to
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 .