Fluctuationlessness approximation and its applications on the remainder term of Taylor expansion: From scratch to present status

Abstract

The general expression of the Fluctuationlessness Theorem states that the matrix representation of an algebraic operator which multiplies its argument by a scalar univariate function, is identical to the image of the independant variable's matrix representation over the same subspace via the same basis set, under that univariate function, when  the fluctuation terms are ignored. Just by using this basic idea, function approximation or numerical quadratures can be constructed. Furthermore this principle applied on the remainder term of a Taylor expansion a highly versatile approximation can be obtained. This review article is just about this approximation aspect of the Fluctuationlessness Theorem.