Pincherle derivative
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Pincherle derivative T' of a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
T: \mathbb \to \mathbb /math> on the
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s in the variable ''x'' over a field \mathbb is the
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 The commutator of two elements, ...
of T with the multiplication by ''x'' in the algebra of endomorphisms \operatorname(\mathbb . That is, T' is another linear operator T': \mathbb \to \mathbb /math> :T' := ,x= Tx-xT = -\operatorname(x)T,\, (for the origin of the \operatorname notation, see the article on the
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is ...
) so that :T'\=T\-xT\\qquad\forall p(x)\in \mathbb This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936).


Properties

The Pincherle derivative, like any
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 The commutator of two elements, ...
, is a
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
, meaning it satisfies the sum and products rules: given two
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s S and T belonging to \operatorname\left( \mathbb \right), #(T + S)^\prime = T^\prime + S^\prime; #(TS)^\prime = T^\prime\!S + TS^\prime where TS = T \circ S is the composition of operators. One also has ,S = ^, S+ , S^/math> where ,S= TS - ST is the usual Lie bracket, which follows from the
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
. The usual
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
, ''D'' = ''d''/''dx'', is an operator on polynomials. By straightforward computation, its Pincherle derivative is : D'= \left(\right)' = \operatorname_ = 1. This formula generalizes to : (D^n)'= \left(\right)' = nD^, by induction. This proves that the Pincherle derivative of a
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 that accepts a function and retur ...
: \partial = \sum a_n = \sum a_n D^n is also a differential operator, so that the Pincherle derivative is a derivation of \operatorname(\mathbb K . When \mathbb has characteristic zero, the shift operator : S_h(f)(x) = f(x+h) \, can be written as : S_h = \sum_ D^n by the Taylor formula. Its Pincherle derivative is then : S_h' = \sum_ D^ = h \cdot S_h. In other words, the shift operators are
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s of the Pincherle derivative, whose spectrum is the whole space of scalars \mathbb. If ''T'' is shift-equivariant, that is, if ''T'' commutes with ''S''''h'' or ,S_h= 0, then we also have ',S_h= 0, so that T' is also shift-equivariant and for the same shift h. The "discrete-time delta operator" : (\delta f)(x) = is the operator : \delta = (S_h - 1), whose Pincherle derivative is the shift operator \delta' = S_h.


See also

*
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 The commutator of two elements, ...
* Delta operator *
Umbral calculus In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blis ...


References

{{Reflist


External links

*Weisstein, Eric W. "
Pincherle Derivative
'". From MathWorld—A Wolfram Web Resource. *

' at the
MacTutor History of Mathematics archive The 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. It contains detailed biographies on many historical and contemporary mathem ...
. Differential algebra