Pincherle derivative

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In mathematics, the Pincherle derivative 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]

so that

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 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 .

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 .

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