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Pincherle DerivativeIn mathematics, the Pincherle derivative of a linear operator T on the space of polynomials in x is another linear operator T′ defined by -
which means that for any polynomial f(x), -
This is a derivation satisfying the sum and product rules: (T + S)′ = T′ + S′ and (TS)′ = T′S + TS′, where TS is the composition of T and S. If T is shift-equivariant, then so is T′. Every shift-equivariant operator on polynomials is of the form -
where D is differentiation with respect to x. When an operator is written in this form, then it is easy to find its Pincherle derivative in this form, by using the fact that -
which may be proved by mathematical induction.
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