Pincherle derivative

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 \operatorname{End}\left( \mathbb{K}[x] \right),

1. (T + S)^\prime = T^\prime + S^\prime;
2. (TS)^\prime = T^\prime!S + TS^\prime where TS = T \circ S is the composition of operators.

One also has [T,S]^{\prime} = [T^{\prime}, S] + [T, S^{\prime}] where [T,S] = TS - ST 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'= \left({d \over {dx}}\right)' = \operatorname{Id}_{\mathbb K [x]} = 1.

This formula generalizes to

: (D^n)'= \left(\right)' = nD^{n-1},

by induction. This 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 \operatorname{Diff}(\mathbb K [x]).

When \mathbb{K} has characteristic zero, the shift operator

: S_h(f)(x) = f(x+h) ,

can be written as

: S_h = \sum_{n \ge 0} {{h^n} \over {n!} }D^n

by the Taylor formula. Its Pincherle derivative is then

: S_h' = \sum_{n \ge 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 \mathbb{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"

: (\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 \delta' = S_h.

References

References

1. (1970). "Graph Theory and Its Applications". Academic Press.