Spectral Integrals of Bernoulli Generalized Functionals

TL;DR

This paper develops a framework for defining and analyzing spectral integrals of Bernoulli generalized functionals within a Gelfand triple, introducing new concepts and demonstrating their properties with examples.

Contribution

It introduces a novel approach to spectral integrals of Bernoulli generalized functionals, expanding the mathematical tools available in this area.

Findings

Established fundamental properties of spectral integrals in this context

Introduced new notions related to Bernoulli generalized functionals

Provided illustrative examples demonstrating the theory

Abstract

Let $\mathcal{S}\subset \mathcal{L}^2 \subset \mathcal{S}^*$ be the Gel'fand triple over the Bernoulli space, where elements of $\mathcal{S}^*$ are called Bernoulli generalized functionals. In this paper, we define integrals of Bernoulli generalized functionals with respect to a spectral measure (projection operator-valued measure) in the framework of $\mathcal{S}\subset \mathcal{L}^2 \subset \mathcal{S}^*$, and examine their fundamental properties. New notions are introduced, several results are obtained and examples are also shown.