Formal distribution

In mathematics, a formal distribution is an infinite sum of powers of a formal variable, usually denoted in the theory of formal distributions. The coefficients of these infinite sums can be many different mathematical structures, such as vector spaces or rings, but in applications most often take values in an algebra over a field. These infinite sums are allowed to have infinitely many positive and negative powers, and are not required to converge, and so do not define functions of the formal variable. Rather, they are interpreted as distributions, that is, linear functionals on an appropriate space of test functions. They are closely related to formal Laurent series, but are not required to have finitely many negative powers. In particular, this means even if the coefficients are ring-valued, it is not necessarily possible to multiply two formal distributions. They are important in the study of vertex operator algebras, since the vertex operator playing a central role in the theory takes values in a space of endomorphism-valued formal distributions. Let be an algebra over , as is the case for applications to vertex algebras. An -valued formal distribution in variables is an arbitrary series with each . These series form a vector space, denoted . While it can be possible to multiply some pairs of elements in the space of formal distributions, in general there is no product on the whole space. In practice, the number of variables considered is often just one or two. If the variables in two formal distributions are disjoint, then the product is well-defined. The product of a formal distribution by a Laurent polynomial is also well-defined. For this section we consider . The formal residue is a linear map , given by The formal residue of can also be written or . It is named after residues from complex analysis, and when is a meromorphic function on a neighborhood of zero in the complex plane, the two notions coincide. The formal derivative is a linear map . For an element , its action is given by extended linearly to give a map for the whole space.