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Pincherle derivative

{{Short description|Type of derivative of a linear operator}}

{{more references|date=June 2013}}

In mathematics, the Pincherle derivative{{cite book |last1=Rota |first1=Gian-Carlo |last2=Mullin |first2=Ronald |title=Graph Theory and Its Applications |date=1970 |publisher=Academic Press |isbn=0123268508 |pages=[https://archive.org/details/graphtheoryitsap0000adva/page/192 192] |url-access=registration |url=https://archive.org/details/graphtheoryitsap0000adva/page/192 }} T' of a linear operator T: \mathbb{K}[x] \to \mathbb{K}[x] on the vector space of polynomials in the variable x over a field \mathbb{K} is the commutator of T with the multiplication by x in the algebra of endomorphisms \operatorname{End}(\mathbb{K}[x]). That is, T' is another linear operator T': \mathbb{K}[x] \to \mathbb{K}[x]

:T' := [T,x] = Tx-xT = -\operatorname{ad}(x)T,\,

(for the origin of the \operatorname{ad} notation, see the article on the adjoint representation) so that

:T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad\forall p(x)\in \mathbb{K}[x].

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 S and T belonging to \operatorname{End}\left( \mathbb{K}[x] \right),

  1. (T + S)^\prime = T^\prime + S^\prime;
  2. (TS)^\prime = T^\prime\!S + TS^\prime where TS = T \circ S is the composition of operators.

One also has [T,S]^{\prime} = [T^{\prime}, S] + [T, S^{\prime}] where [T,S] = TS - ST 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

: D'= \left({d \over {dx}}\right)' = \operatorname{Id}_{\mathbb K [x]} = 1.

This formula generalizes to

: (D^n)'= \left({{d^n} \over {dx^n}}\right)' = nD^{n-1},

by induction. This proves that the Pincherle derivative of a differential operator

: \partial = \sum a_n {{d^n} \over {dx^n} } = \sum a_n D^n

is also a differential operator, so that the Pincherle derivative is a derivation of \operatorname{Diff}(\mathbb K [x]).

When \mathbb{K} has characteristic zero, the shift operator

: S_h(f)(x) = f(x+h) \,

can be written as

: S_h = \sum_{n \ge 0} {{h^n} \over {n!} }D^n

by the Taylor formula. Its Pincherle derivative is then

: S_h' = \sum_{n \ge 1} {{h^n} \over {(n-1)!} }D^{n-1} = h \cdot S_h.

In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars \mathbb{K}.

If T is shift-equivariant, that is, if T commutes with Sh or [T,S_h] = 0, then we also have [T',S_h] = 0, so that T' is also shift-equivariant and for the same shift h.

The "discrete-time delta operator"

: (\delta f)(x) = {{ f(x+h) - f(x) } \over h }

is the operator

: \delta = {1 \over h} (S_h - 1),

whose Pincherle derivative is the shift operator \delta' = S_h.

See also

References

{{Reflist}}

External links

  • Weisstein, Eric W. "[http://mathworld.wolfram.com/PincherleDerivative.html Pincherle Derivative]". From MathWorld—A Wolfram Web Resource.
  • [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Pincherle.html Biography of Salvatore Pincherle] at the MacTutor History of Mathematics archive.

Category:Differential algebra