Pincherle derivative

In mathematics, the Pincherle derivative T’ of a linear operator T:K[x] → K[x] on the vector space of polynomials in the variable x over a field K is the commutator of T with the multiplication by x in the algebra of endomorphisms End(K[x]). That is, T’ is another linear operator T’:K[x] → K[x]

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

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

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

One also has $\scriptstyle {[T,S]^{\prime }=[T^{\prime },S]+[T,S^{\prime }]}$ where $\scriptstyle {[T,S]=TS-ST}$ 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. It 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 $\scriptstyle \operatorname {Diff}({\mathbb K}[x])$.

The shift operator

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

can be written as

$S_{h}=\sum _{{n=0}}{{h^{n}} \over {n!}}D^{n}$

by the Taylor formula. Its Pincherle derivative is then

$S_{h}'=\sum _{{n=1}}{{h^{n}} \over {(n-1)!}}D^{{n-1}}=h\cdot S_{h}.$

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

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

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 $\scriptstyle {\delta '=S_{h}}$.

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.