Pincherle derivative

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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 $\scriptstyle{ T':\mathbb K[x] \longrightarrow \mathbb K[x] }$ defined as

$T' = [T,x] = Tx-xT = -ad(x)T,\,$

so that

$T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad\forall p(x)\in \mathbb{K}[x].$

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

This concept is named after the Italian mathematician Salvatore Pincherle (1853—1936).

[edit] 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 $\scriptstyle End \left( \mathbb K[x] \right)$

$\scriptstyle{ (T + S)^\prime = T^\prime + S^\prime }$ ;
$\scriptstyle{ (TS)^\prime = T^\prime\!S + TS^\prime }$ 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 $\scriptstyle{D'= ({d \over {dx}})' = Id_{\mathbb K [x]}} = 1$ .

This formula generalizes to $\scriptstyle{(D^n)'=({{d^n} \over {dx^n}})'=nD^{n-1}}$ , 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 $\scriptstyle{S_h' = \sum_n {{h^n} \over {(n-1)!} }D^{n-1} = h \cdot S_h}$ . 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 $\scriptstyle{ [T',S_h] = 0}$ , so that $\scriptstyle T'$ 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 $\scriptstyle{ \delta ' = S_h }$ .