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

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

so that

:$T\text{'}\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}qquadforall\; p(x)in\; mathbb\{K\}\; [x]\; .$

In other words, Pincherle derivation is the commutator of $scriptstyle\{T\}$ with the multiplication by $scriptstyle\; x$ in the algebra of endomorphisms $scriptstyle\{\; operatorname\{End\}\; left(\; mathbb\; K\; [x]\; ight)\; \}$.

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 $scriptstyle\; operatorname\{End\}\; left(\; mathbb\; K\; [x]\; ight)$

#$scriptstyle\{\; (T\; +\; S)^prime\; =\; T^prime\; +\; S^prime\; \}$ ;
#$scriptstyle\{\; (TS)^prime\; =\; T^prime!S\; +\; TS^prime\; \}$ 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, "D" = "d"/"dx", is an operator on polynomials. By straightforward computation, its Pincherle derivative is

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

This formula generalizes to

: $(D^n)\text{'}=\; left($d^n} over {dx^n ight)' = 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

: $S\_h\text{'}\; =\; sum\_\{n=1\}$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 $scriptstyle\{\; [T\text{'},S\_h]\; =\; 0\}$, so that $scriptstyle\; T\text{'}$ is also shift-equivariant and for the same shift $scriptstyle\; h$.

The "discrete-time delta operator"

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

is the operator

: $delta\; =\; \{1\; over\; h\}\; (S\_h\; -\; 1),$

whose Pincherle derivative is the shift operator $scriptstyle\{\; delta\; \text{'}\; =\; S\_h\; \}$.

See also

*Commutator
*Delta operator
*Umbral calculus

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.