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':mathbb K [x] longrightarrow mathbb K [x] } defined as

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

so that

:$$T'{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'= left({d over {dx ight)' = operatorname{Id}_{mathbb K [x] } = 1.

This formula generalizes to

: $$ (D^n)'= 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' = 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. " [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.