词条 | Pincherle derivative |
释义 |
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] so that This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936). PropertiesThe Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators and belonging to
One also has where 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 This formula generalizes to by induction. It proves that the Pincherle derivative of a differential operator is also a differential operator, so that the Pincherle derivative is a derivation of . The shift operator can be written as by the Taylor formula. Its Pincherle derivative is then In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars . If T is shift-equivariant, that is, if T commutes with Sh or , then we also have , so that is also shift-equivariant and for the same shift . The "discrete-time delta operator" is the operator whose Pincherle derivative is the shift operator . See also
External links
1 : Differential algebra |
随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识,基本涵盖了大多数领域的百科知识,是一部内容自由、开放的电子版国际百科全书。