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Pincherle derivative

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Type of derivative of a linear operator
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In mathematics, the Pincherle derivative T ′ {\displaystyle T'} of a linear operator T : K [ x ] → K [ x ] {\displaystyle T:\mathbb {K} \to \mathbb {K} } on the vector space of polynomials in the variable x over a field K {\displaystyle \mathbb {K} } is the commutator of T {\displaystyle T} with the multiplication by x in the algebra of endomorphisms End ⁡ ( K [ x ] ) {\displaystyle \operatorname {End} (\mathbb {K} )} . That is, T ′ {\displaystyle T'} is another linear operator T ′ : K [ x ] → K [ x ] {\displaystyle T':\mathbb {K} \to \mathbb {K} }

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

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

T ′ { p ( x ) } = T { x p ( x ) } − x T { p ( x ) } ∀ p ( x ) ∈ K [ x ] . {\displaystyle T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad \forall p(x)\in \mathbb {K} .}

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

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

One also has [ T , S ] ′ = [ T ′ , S ] + [ T , S ′ ] {\displaystyle ^{\prime }=+} where [ T , S ] = T S − S T {\displaystyle =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 ′ = ( d d x ) ′ = Id K [ x ] = 1. {\displaystyle D'=\left({d \over {dx}}\right)'=\operatorname {Id} _{\mathbb {K} }=1.}

This formula generalizes to

( D n ) ′ = ( d n d x n ) ′ = n D n − 1 , {\displaystyle (D^{n})'=\left({{d^{n}} \over {dx^{n}}}\right)'=nD^{n-1},}

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

∂ = ∑ a n d n d x n = ∑ a n D n {\displaystyle \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 Diff ⁡ ( K [ x ] ) {\displaystyle \operatorname {Diff} (\mathbb {K} )} .

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

S h ( f ) ( x ) = f ( x + h ) {\displaystyle S_{h}(f)(x)=f(x+h)\,}

can be written as

S h = ∑ n ≥ 0 h n n ! D n {\displaystyle S_{h}=\sum _{n\geq 0}{{h^{n}} \over {n!}}D^{n}}

by the Taylor formula. Its Pincherle derivative is then

S h ′ = ∑ n ≥ 1 h n ( n − 1 ) ! D n − 1 = h ⋅ S h . {\displaystyle S_{h}'=\sum _{n\geq 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 K {\displaystyle \mathbb {K} } .

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

The "discrete-time delta operator"

( δ f ) ( x ) = f ( x + h ) − f ( x ) h {\displaystyle (\delta f)(x)={{f(x+h)-f(x)} \over h}}

is the operator

δ = 1 h ( S h − 1 ) , {\displaystyle \delta ={1 \over h}(S_{h}-1),}

whose Pincherle derivative is the shift operator δ ′ = S h {\displaystyle \delta '=S_{h}} .

See also

References

  1. Rota, Gian-Carlo; Mullin, Ronald (1970). Graph Theory and Its Applications. Academic Press. pp. 192. ISBN 0123268508.

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