Approximation principles of functionals by linear functionals [Lecture]

Almost multiplicative linear functionals and approximate spectrum

We define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital Banach algebra A and show that the δ-approximate spectrum σ_δ (a) of a is compact. The relation between the δ-approximate spectrum and the usual spectrum is investigated. Also an analogue of the classical Gleason-Kahane-Zelazko theorem is established: For each ε>0, there is δ>0 such that if ϕ is...

Ordering Points by Linear Functionals

Given a set of points in Euclidean space, we say that two linear functionals diier on that set if they give rise to diierent linear orderings of the points. We investigate what the largest and smallest number of diierent linear functionals can be as a function of the number of points and the dimension of the space. 1. Introduction The purpose of this paper is to investigate the number of diiere...

Symmetries of Linear Functionals

It is shown that a linear functional on a space of functions can be described by G, a group of its symmetries, together with the restriction of to certain G-invariant functions. This simple consequence of invariant theory has long been used, implicitly, in the construction of numerical integration rules. It is the author's hope that, by showing that these ideas have nothing to do with the origi...

ON APPROXIMATION OF LINEAR FUNCTIONALS ON Lp SPACES

In a recent paper certain approximations to continuous nonlinear functionals de-ned on an L p space (1 < p < 1) are shown to exist. These approximations may be realized by sigmoidal neural networks employing a linear input layer that implements nite sums of integrals of a certain type. In another recent paper similar approximation results are obtained using elements of a general class of contin...

Approximation of Non-convex Functionals in Gbv Approximation of Non-convex Functionals in Gbv