Power Integral Representation for a Kronecker Power Series to a Multivariate Function and Its Coefficients

摘要

This paper focuses on the power integral representation of the Kronecker power series coefficients. A multivariate function can be represented as an infinite series of scalars each of which is a product of an appropriate vector transpose and a compatible Kronecker power of the state vector whose elements are composed of the independent variable including first degree expressions. The main idea is to state the each vector transpose coefficient as the integral of a vector's appropriate Kronecker power multiplied by a common factor which depends on the integration variables. We can show that the common function factor can always be uniquely determined from the Kronecker power series. However, it can not be guaranteed to be a weight function unless the focused Kronecker power series representation coefficients fulfill certain conditions. We do not focus on the weight function related issues here.