Pincherle derivative
Encyclopedia
In mathematics
, the Pincherle derivative of a linear operator
on the vector space
of polynomial
s in the variable
over a field
is another linear operator 
defined as
so that
In other words, Pincherle derivation is the commutator
of
with the multiplication by 
in the algebra of endomorphisms
.
This concept is named after the Italian mathematician Salvatore Pincherle
(1853–1936).
, is a derivation
, meaning it satisfies the sum and products rules: given two linear operators
and 
belonging to 
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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Pincherle derivative of a linear operator
on the vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
of polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s in the variable
over a fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
is another linear operator defined asso that
In other words, Pincherle derivation is the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
of
with the multiplication by in the algebra of endomorphismsEndomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
.This concept is named after the Italian mathematician Salvatore Pincherle
Salvatore Pincherle
Salvatore Pincherle was an Italian mathematician. He contributed significantly to the field of functional analysis, established the Italian Mathematical Union , and was president of the Third International Congress of Mathematicians...
(1853–1936).
Properties
The Pincherle derivative, like any commutatorCommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
, is a derivation
Derivation (abstract algebra)
In abstract algebra, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D: A → A that satisfies Leibniz's law: D = b + a.More...
, meaning it satisfies the sum and products rules: given two linear operators
and belonging to 
;
where 
is the composition of operators ;
where 
is the usual Lie bracketLie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
.
The usual derivative, D = d/dx, is an operator on polynomials. By straightforward computation, its Pincherle derivative is
This formula generalizes to
by induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
. It proves that the Pincherle derivative of a differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
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
.External links
- Weisstein, Eric W. "Pincherle Derivative". From MathWorld--A Wolfram Web Resource.
- Biography of Salvatore Pincherle at the MacTutor History of Mathematics archiveMacTutor History of Mathematics archiveThe MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland...
.