Pincherle derivative
Template:Short description Template:More references In mathematics, the Pincherle derivative<ref>Template:Cite book</ref> <math>T'</math> of a linear operator <math>T: \mathbb{K}[x] \to \mathbb{K}[x]</math> on the vector space of polynomials in the variable x over a field <math>\mathbb{K}</math> is the commutator of <math>T</math> with the multiplication by x in the algebra of endomorphisms <math>\operatorname{End}(\mathbb{K}[x])</math>. That is, <math>T'</math> is another linear operator <math>T': \mathbb{K}[x] \to \mathbb{K}[x]</math>
- <math>T' := [T,x] = Tx-xT = -\operatorname{ad}(x)T,\,</math>
(for the origin of the <math>\operatorname{ad}</math> notation, see the article on the adjoint representation) so that
- <math>T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad\forall p(x)\in \mathbb{K}[x].</math>
This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936).
Properties
The Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators <math>S</math> and <math>T</math> belonging to <math>\operatorname{End}\left( \mathbb{K}[x] \right),</math>
- <math>(T + S)^\prime = T^\prime + S^\prime</math>;
- <math>(TS)^\prime = T^\prime\!S + TS^\prime</math> where <math>TS = T \circ S</math> is the composition of operators.
One also has <math>[T,S]^{\prime} = [T^{\prime}, S] + [T, S^{\prime}]</math> where <math>[T,S] = TS - ST</math> 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
- <math>D'= \left({d \over {dx}}\right)' = \operatorname{Id}_{\mathbb K [x]} = 1.</math>
This formula generalizes to
- <math>(D^n)'= \left({{d^n} \over {dx^n}}\right)' = nD^{n-1},</math>
by induction. This proves that the Pincherle derivative of a differential operator
- <math>\partial = \sum a_n {{d^n} \over {dx^n} } = \sum a_n D^n</math>
is also a differential operator, so that the Pincherle derivative is a derivation of <math>\operatorname{Diff}(\mathbb K [x])</math>.
When <math>\mathbb{K}</math> has characteristic zero, the shift operator
- <math>S_h(f)(x) = f(x+h) \,</math>
can be written as
- <math>S_h = \sum_{n \ge 0} {{h^n} \over {n!} }D^n</math>
by the Taylor formula. Its Pincherle derivative is then
- <math>S_h' = \sum_{n \ge 1} {{h^n} \over {(n-1)!} }D^{n-1} = h \cdot S_h.</math>
In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars <math>\mathbb{K}</math>.
If T is shift-equivariant, that is, if T commutes with Sh or <math>[T,S_h] = 0</math>, then we also have <math>[T',S_h] = 0</math>, so that <math>T'</math> is also shift-equivariant and for the same shift <math>h</math>.
The "discrete-time delta operator"
- <math>(\delta f)(x) = {{ f(x+h) - f(x) } \over h }</math>
is the operator
- <math>\delta = {1 \over h} (S_h - 1),</math>
whose Pincherle derivative is the shift operator <math>\delta' = S_h</math>.
See also
References
External links
- Weisstein, Eric W. "Pincherle Derivative". From MathWorld—A Wolfram Web Resource.
- Biography of Salvatore Pincherle at the MacTutor History of Mathematics archive.