Positive definite functionals in Banach spaces

Positive Definite Functionals in Banach Spaces

We establish a version of Bochner Theorem due to S. Boylan for Banach spaces with a basis.

Minimization of Tikhonov Functionals in Banach Spaces

Tikhonov functionals are known to be well suited for obtaining regularized solutions of linear operator equations. We analyze two iterative methods for finding the minimizer of norm-based Tikhonov functionals in Banach spaces. One is the steepest descent method, whereby the iterations are directly carried out in the underlying space, and the other one performs iterations in the dual space. We p...

Linear Functionals on Certain Banach Spaces

The Gelfand transform, positive linear functionals, and positive-definite functions

In this note, unless we say otherwise every vector space or algebra we speak about is over C. If A is a Banach algebra and e ∈ A satisfies xe = x and ex = x for all x ∈ A, and also ‖e‖ = 1, we say that e is unity and that A is unital. If A is a unital Banach algebra and x ∈ A, the spectrum of x is the set σ(x) of those λ ∈ C for which λe−x is not invertible. It is a fact that if A is a unital B...

Metric Spaces and Positive Definite

As p—► » we get the space E„ with the distance function max,-=i,... ,m | x< — as/ |. Let, furthermore, lp stand for the space of real sequences with the series of pth powers of the absolute values convergent. Similarly let Lp denote the space of real measurable functions in the interval (0, 1) which are oummable to the ^»th power, while C shall mean the space of real continuous functions in the...