Pincherle derivative

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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 Sh 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

External links

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