Pincherle derivative

Short description: Type of derivative of a linear operator

In mathematics, the Pincherle derivative $T^{\prime }$ 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 $\mathrm{End}(\mathbb{𝕂}[x])$. That is, $T^{\prime }$ is another linear operator $T^{\prime }:\mathbb{𝕂}[x]\to \mathbb{𝕂}[x]$

$T^{\prime }:=[T,x]=Tx-xT=-\mathrm{ad}(x)T,$

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

$T^{\prime }\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\mathrm{\forall }p(x)\in \mathbb{𝕂}[x].$

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 $\mathrm{End}(\mathbb{𝕂}[x]),$

1. $(T+S{)}^{\prime }=T^{\prime }+S^{\prime }$;
2. $(TS{)}^{\prime }=T^{\prime }S+T{S}^{\prime }$ 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

$D^{\prime }=\left(\frac{d}{dx}\right)={\mathrm{Id}}_{\mathbb{𝕂}[x]}=1.$

This formula generalizes to

$(D^n{)}^{\prime }=\left(\frac{d^n}{d{x}^n}\right)=n{D}^{n-1},$

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

$\partial =\sum a_n\frac{d^n}{d{x}^n}=\sum a_n{D}^n$

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

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

$S_h(f)(x)=f(x+h)$

can be written as

$S_h=\sum _{n\ge 0}\frac{h^n}{n!}{D}^n$

by the Taylor formula. Its Pincherle derivative is then

${S_h}^{\prime }=\sum _{n\ge 1}\frac{h^n}{(n-1)!}{D}^{n-1}=h\cdot S_h.$

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 S_h or $[T,S_h]=0$, then we also have $[T^{\prime },S_h]=0$, so that $T^{\prime }$ is also shift-equivariant and for the same shift $h$.

The "discrete-time delta operator"

$(\delta f)(x)=\frac{f(x+h)-f(x)}{h}$

is the operator

$\delta =\frac{1}{h}(S_h-1),$

whose Pincherle derivative is the shift operator ${\delta }^{\prime }=S_h$.

See also

• Commutator
• Delta operator
• Umbral calculus

External links

• Weisstein, Eric W. "Pincherle Derivative". From MathWorld—A Wolfram Web Resource.
• Biography of Salvatore Pincherle at the MacTutor History of Mathematics archive.