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| | Individual who wrote the is called Eusebio. South Carolina is its birth place. The favorite hobby for him and moreover his kids is in order to fish and he's been really doing it for quite some time. Filing has been his profession although. Go to his website to identify a out more: http://prometeu.net<br><br>my webpage: [http://prometeu.net clash of clans hack] |
| In [[mathematics]], the '''Pincherle derivative''' ''T’'' of a [[linear operator]] ''T'':'''K'''[''x''] → '''K'''[''x''] on the [[vector space]] of [[polynomial]]s in the variable ''x'' over a [[field (mathematics)|field]] '''K''' is the [[commutator]] of ''T'' with the multiplication by ''x'' in the [[endomorphism ring|algebra of endomorphisms]] End('''K'''[''x'']). That is, ''T’'' is another linear operator ''T’'':'''K'''[''x''] → '''K'''[''x'']
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| :<math> T' := [T,x] = Tx-xT = -\operatorname{ad}(x)T,\,</math>
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| so that
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| :<math> T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad\forall p(x)\in \mathbb{K}[x].</math>
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| This concept is named after the Italian mathematician [[Salvatore Pincherle]] (1853–1936).
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| == Properties ==
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| The Pincherle derivative, like any [[commutator]], is a [[derivation (abstract algebra)|derivation]], meaning it satisfies the sum and products rules: given two [[linear operator]]s <math>\scriptstyle S </math> and <math>\scriptstyle T </math> belonging to <math> \scriptstyle \operatorname{End} \left( \mathbb K[x] \right) </math> | |
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| #<math>\scriptstyle{ (T + S)^\prime = T^\prime + S^\prime }</math> ;
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| #<math>\scriptstyle{ (TS)^\prime = T^\prime\!S + TS^\prime }</math> where <math>\scriptstyle{ TS = T \circ S}</math> is the [[Function_composition#Composition_operator|composition of operators]] ;
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| One also has <math>\scriptstyle{ [T,S]^\prime = [T^\prime , S] + [T, S^\prime ] }</math> where <math>\scriptstyle{ [T,S] = TS - ST}</math> is the usual [[Lie algebra|Lie bracket]], which follows from the [[Jacobi identity]].
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| The usual derivative, ''D'' = ''d''/''dx'', is an operator on polynomials. By straightforward computation, its Pincherle derivative is
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| : <math> D'= \left({d \over {dx}}\right)' = \operatorname{Id}_{\mathbb K [x]} = 1.</math>
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| This formula generalizes to
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| : <math> (D^n)'= \left({{d^n} \over {dx^n}}\right)' = nD^{n-1},</math>
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| by [[mathematical induction|induction]]. It proves that the Pincherle derivative of a [[differential operator]]
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| : <math> \partial = \sum a_n {{d^n} \over {dx^n} } = \sum a_n D^n </math> | |
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| is also a differential operator, so that the Pincherle derivative is a derivation of <math>\scriptstyle \operatorname{Diff}(\mathbb K [x]) </math>.
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| The shift operator
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| : <math> S_h(f)(x) = f(x+h) \, </math> | |
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| can be written as
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| : <math> S_h = \sum_{n=0} {{h^n} \over {n!} }D^n </math>
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| by the [[Taylor formula]]. Its Pincherle derivative is then
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| : <math> S_h' = \sum_{n=1} {{h^n} \over {(n-1)!} }D^{n-1} = h \cdot S_h. </math>
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| In other words, the shift operators are [[eigenvector]]s of the Pincherle derivative, whose spectrum is the whole space of scalars <math>\scriptstyle{ \mathbb K }</math>.
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| If ''T'' is [[shift-equivariant]], that is, if ''T'' commutes with ''S''<sub>''h''</sub> or <math>\scriptstyle{ [T,S_h] = 0}</math>, then we also have <math>\scriptstyle{ [T',S_h] = 0}</math>, so that <math>\scriptstyle T'</math> is also shift-equivariant and for the same shift <math>\scriptstyle h</math>.
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| The "discrete-time delta operator"
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| : <math> (\delta f)(x) = {{ f(x+h) - f(x) } \over h }</math> | |
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| is the operator
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| : <math> \delta = {1 \over h} (S_h - 1),</math>
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| whose Pincherle derivative is the shift operator <math>\scriptstyle{ \delta ' = S_h }</math>.
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| == See also ==
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| *[[Commutator]]
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| *[[Delta operator]]
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| *[[Umbral calculus]]
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| == External links ==
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| *Weisstein, Eric W. "''[http://mathworld.wolfram.com/PincherleDerivative.html Pincherle Derivative]''". From MathWorld—A Wolfram Web Resource.
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| *''[http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Pincherle.html Biography of Salvatore Pincherle]'' at the [[MacTutor History of Mathematics archive]].
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| [[Category:Differential algebra]]
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Individual who wrote the is called Eusebio. South Carolina is its birth place. The favorite hobby for him and moreover his kids is in order to fish and he's been really doing it for quite some time. Filing has been his profession although. Go to his website to identify a out more: http://prometeu.net
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