Pincherle derivative - Alchetron, The Free Social Encyclopedia

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 ] = T x − x T = − ad ⁡ ( x ) T ,

so that

T ′ { p ( x ) } = T { x p ( x ) } − x T { p ( x ) } ∀ p ( x ) ∈ 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 S and T belonging to End ⁡ ( K [ x ] )

( T + S ) ′ = T ′ + S ′ ;
( T S ) ′ = T ′ S + T S ′ where T S = T ∘ S is the composition of operators ;

One also has [ T , S ] ′ = [ T ′ , S ] + [ T , S ′ ] where [ T , S ] = T S − S T 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 ′ = ( d d x ) ′ = Id K [ x ] = 1.

This formula generalizes to

( D n ) ′ = ( d n d x n ) ′ = n D n − 1 ,

by induction. It proves that the Pincherle derivative of a differential operator

∂ = ∑ a n d n d x n = ∑ a n D n

is also a differential operator, so that the Pincherle derivative is a derivation of Diff ⁡ ( K [ x ] ) .

The shift operator

S h ( f ) ( x ) = f ( x + h )

can be written as

S h = ∑ n = 0 h n n ! D n

by the Taylor formula. Its Pincherle derivative is then

S h ′ = ∑ n = 1 h n ( n − 1 ) ! D n − 1 = h ⋅ S h .

In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars K .

If T is shift-equivariant, that is, if T commutes with S_h or [ T , S h ] = 0 , then we also have [ T ′ , S h ] = 0 , so that T ′ is also shift-equivariant and for the same shift h .

The "discrete-time delta operator"

( δ f ) ( x ) = f ( x + h ) − f ( x ) h

is the operator

δ = 1 h ( S h − 1 ) ,

whose Pincherle derivative is the shift operator δ ′ = S h .

References

Pincherle derivative Wikipedia

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