NationMaster - Encyclopedia: 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 For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

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

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 $scriptstyle{ End left( mathbb K[x] right) }$ . In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ... In abstract algebra, one associates to certain objects a ring, the objects endomorphism ring, which encodes several internal properties of the object. ...

This concept is named after the Italian mathematician Salvatore Pincherle (1853—1936). Salvatore Pincherle (February 11, 1853 â€” July 19, 1936) was an Italian mathematician. ...

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 $scriptstyle End left( mathbb K[x] right)$ In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

$scriptstyle{ (T + S)^prime = T^prime + S^prime }$ ;
$scriptstyle{ (TS)^prime = T^prime!S + TS^prime }$ 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 $scriptstyle{D'= ({d over {dx}})' = Id_{mathbb K [x]}} = 1$ . In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

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 $scriptstyle{ Diff(mathbb K [x]) }$ . Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... In mathematics, a differential operator is an operator defined as a function of the differentiation operator. ...

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 $scriptstyle{ mathbb K }$ . As the degree of the taylor series rises, it approaches the correct function. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

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 $scriptstyle h$ . In mathematics, a delta operator is a shift-equivariant linear operator Q on the vector space of polynomials in a variable x that reduces degrees by one. ...

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 $scriptstyle{ delta ' = S_h }$ .

External links

Weisstein, Eric W. "Pincherle Derivative". From MathWorld--A Wolfram Web Resource.
Biography of Salvatore Pincherle at the MacTutor History of Mathematics archive.

Categories: Differential algebra

The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ...

Results from FactBites:

derivative - Article and Reference from OnPedia.com (2086 words)
	In mathematics, the derivative of a function is one of the two central concepts of calculus.
	If the second derivative is positive at a critical point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum.
	Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.

Derivation - Wikinfo (131 words)
	Examples of derivations are partial derivatives, Lie derivatives, the Pincherle derivative, and the commutator with respect to an element of the algebra.
	All these examples are tightly related, with the concept of derivation as the major unifying theme.
	Derivation may also be used as a synonym for proof, particularly for formulae.

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