Lexicographic Effect Algebras

In the paper we investigate a class of effect algebras which can be represented in the form of the lexicographic product \Gamma(H\lex G,(u,0)), where $(H,u)$ is an Abelian unital po-group and $G$ is an Abelian directed po-group. We study algebraic conditions when an effect algebra is of this form. Fixing a unital po-group $(H,u)$, the category of strong $(H,u)$-perfect effect algebra is introduced and it is shown that it is categorically equivalent to the category of directed po-group with interpolation. We show some representation theorems including a subdirect product representation by antilattice lexicographic effect algebras

Representation of States on Effect-Tribes and Effect Algebras by Integrals

We describe $\sigma$-additive states on effect-tribes by integrals. Effect-tribes are monotone $\sigma$-complete effect algebras of functions where operations are defined by points. Then we show that every state on an effect algebra is an integral through a Borel regular probability measure. Finally, we show that every $\sigma$-convex combination of extremal states on a monotone $\sigma$-complete effect algebra is a Jauch-Piron state.Comment: 20 page

Kite Pseudo Effect Algebras

We define a new class of pseudo effect algebras, called kite pseudo effect algebras, which is connected with partially ordered groups not necessarily with strong unit. In such a case, starting even with an Abelian po-group, we can obtain a noncommutative pseudo effect algebra. We show how such kite pseudo effect algebras are tied with different types of the Riesz Decomposition Properties. Kites are so-called perfect pseudo effect algebras, and we define conditions when kite pseudo effect algebras have the least non-trivial normal ideal

Smearing of Observables and Spectral Measures on Quantum Structures

An observable on a quantum structure is any $\sigma$-homomorphism of quantum structures from the Borel $\sigma$-algebra of the real line into the quantum structure which is in our case a monotone $\sigma$-complete effect algebras with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean $\sigma$-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there is its spectral measure

The Dirichlet space: A Survey

In this paper we survey many results on the Dirichlet space of analytic functions. Our focus is more on the classical Dirichlet space on the disc and not the potential generalizations to other domains or several variables. Additionally, we focus mainly on certain function theoretic properties of the Dirichlet space and omit covering the interesting connections between this space and operator theory. The results discussed in this survey show what is known about the Dirichlet space and compares it with the related results for the Hardy space.Comment: 35 pages, typoes corrected, some open problems adde

Sampling Theorem and Discrete Fourier Transform on the Hyperboloid

Using Coherent-State (CS) techniques, we prove a sampling theorem for holomorphic functions on the hyperboloid (or its stereographic projection onto the open unit disk $\mathbb D_1$), seen as a homogeneous space of the pseudo-unitary group SU(1,1). We provide a reconstruction formula for bandlimited functions, through a sinc-type kernel, and a discrete Fourier transform from $N$ samples properly chosen. We also study the case of undersampling of band-unlimited functions and the conditions under which a partial reconstruction from $N$ samples is still possible and the accuracy of the approximation, which tends to be exact in the limit $N\to\infty$.Comment: 22 pages, 2 figures. Final version published in J. Fourier Anal. App