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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]
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 .
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 .