- Pincherle derivative
In
mathematics , the Pincherle derivative of alinear operator on thevector space ofpolynomial s in the variable over a field is another linear operator defined as:
so that
:
In other words, Pincherle derivation is the
commutator of with the multiplication by in the algebra of endomorphisms .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 twolinear operator s and belonging to# ;
# where is the composition of operators ;
# where is the usual Lie bracket.The usual derivative, "D" = "d"/"dx", is an operator on polynomials. By straightforward computation, its Pincherle derivative is
:
This formula generalizes to
:
by induction. It proves that the Pincherle derivative of a
differential operator :
is also a differential operator, so that the Pincherle derivative is a derivation of .
The shift operator
:
can be written as
:
by the
Taylor formula . Its Pincherle derivative is then:
In other words, the shift operators are
eigenvector s of the Pincherle derivative, whose spectrum is the whole space of scalars .If "T" is
shift-equivariant , that is, if "T" commutes with "S""h" or , then we also have , so that is also shift-equivariant and for the same shift .The "discrete-time delta operator"
:
is the operator
:
whose Pincherle derivative is the shift operator .
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 theMacTutor History of Mathematics archive .
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