PINCHERLE DERIVATIVE

In mathematics, the 'Pincherle derivative' of a linear operator $scriptstyle\{\; T:mathbb\; K[x]\; longrightarrow\; mathbb\; K[x]\; \}$ on the vector space of polynomials in the variable $scriptstyle\; x$ over a field $scriptstyle\{\; mathbb\; K\}$ is another linear operator $scriptstyle\{\; T\text{'}:mathbb\; K[x]\; longrightarrow\; mathbb\; K[x]\; \}$ defined as
:$T\text{'}\; =\; [T,x]\; =\; Tx-xT\; =\; -ad(x)T,,$
so that
:
In other words, Pincherle derivation is the commutator of $scriptstyle\{T\}$ with the multiplication by $scriptstyle\; x$ in the algebra of endomorphisms $scriptstyle\{\; End\; left(\; mathbb\; K[x]$
ight) }.
This concept is named after the Italian mathematician Salvatore Pincherle (1853—1936).

Contents

Properties

See also

External links

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 $scriptstyle\; End\; left(\; mathbb\; K[x]$
ight)
#$scriptstyle\{\; (T\; +\; S)^prime\; =\; T^prime\; +\; S^prime\; \}$ ;
#$scriptstyle\{\; (TS)^prime\; =\; T^prime!S\; +\; TS^prime\; \}$ where $scriptstyle\{\; TS\; =\; T\; circ\; S\}$ is the composition of operators ;
#$scriptstyle\{\; [T,S]^prime\; =\; [T^prime\; ,\; S]\; +\; [T,\; S^prime\; ]\; \}$ where $scriptstyle\{\; [T,S]\; =\; TS\; -\; ST\}$ is the usual Lie bracket.
The usual derivative, $scriptstyle\{D\; =\; \{d\; over\; dx\}\; \}$ is an operator on polynomials. By straightforward computation, its Pincherle derivative is $scriptstyle\{D\text{'}=\; (\{d\; over\; \{dx\}\})\text{'}\; =\; Id\_\{mathbb\; K\; [x]\}\}\; =\; 1$.
This formula generalizes to $scriptstyle\{(D^n)\text{'}=(\{\{d^n\}\; over\; \{dx^n\}\})\text{'}=nD^\{n-1\}\}$, by induction. It proves that the Pincherle derivative of a differential operator $scriptstyle\{\; partial\; =\; sum\; a\_n\; \{\{d^n\}\; over\; \{dx^n\}\; \}\; =\; sum\; a\_n\; D^n\; \}$ is also a differential operator, so that the Pincherle derivative is a derivation of $scriptstyle\{\; Diff(mathbb\; K\; [x])\; \}$.
The shift operator $scriptstyle\{S\_h(f)(x)\; =\; f(x+h)\; \}$ can be written as $scriptstyle\{S\_h\; =\; sum\_n\; \{\{h^n\}\; over\; \{n!\}\; \}D^n\; \}$ by the Taylor formula. Its Pincherle derivative is then $scriptstyle\{S\_h\text{'}\; =\; sum\_n\; \{\{h^n\}\; over\; \{(n-1)!\}\; \}D^\{n-1\}\; =\; h\; cdot\; S\_h\}$. In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars $scriptstyle\{\; mathbb\; K\; \}$.
If $scriptstyle\; T$ is shift-equivariant, that is, if $scriptstyle\; T$ commutes with $scriptstyle\; S\_h$ or $scriptstyle\{\; [T,S\_h]\; =\; 0\}$, then we also have $scriptstyle\{\; [T\text{'},S\_h]\; =\; 0\}$, so that $scriptstyle\; T\text{'}$ is also shift-equivariant and for the same shift $scriptstyle\; h$.
The "discrete-time delta operator" $scriptstyle\; \{(delta\; f)(x)\; =\; \{\{\; f(x+h)\; -\; f(x)\; \}\; over\; h\; \}\}$ is the operator $scriptstyle\{\; delta\; =\; \{1\; over\; h\}\; (S\_h\; -\; 1)\}$, whose Pincherle derivative is the shift operator $scriptstyle\{\; delta\; \text{'}\; =\; S\_h\; \}$.

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.