Integrability on the Abstract Wiener Space of Double Sequences and Fernique Theorem

Jeong-Gyoo Kim,Kwok-Pun Ho

doi:10.1155/2021/1667865

Abstract

The integrability of a function defined on the abstract Wiener space of double Fourier coefficients is explored. The abstract Wiener space is also a Hilbert space. We define an orthonormal system of the Hilbert space to establish a measure and integration on the abstract Wiener space. We examine the integrability of a function e α · 2 defined on the abstract Wiener space for Fernique theorem. With respect to the abstract Wiener measure, the integral of the function turns out to be convergent for α < 1 / 2 . The result provides a wider choice of the constant α than that of Fernique.

Highlights

We explore an abstract Wiener space that consists of double Fourier coefficients focused on the integrability of functions defined on the space
We examine the integral of eαk·k2, i.e., Fernique theorem in the abstract Wiener space
We explored the abstract Wiener space U consisting of sequences of double Fourier coefficients for integrability

Summary

Introduction

We explore an abstract Wiener space that consists of double Fourier coefficients focused on the integrability of functions defined on the space. We define an orthonormal system of the Hilbert space and utilise the system to define a probability measure on the abstract Wiener space of double sequences and develop to integration. We examine the integral of eαk·k2 , i.e., Fernique theorem in the abstract Wiener space. An abstract Wiener space of double Fourier coefficients is defined in the author’s work [5] where detailed development of the space can be found. We introduce an abstract Wiener space as well as the Hilbert spaces of double Fourier coefficients .

Preliminaries

An Orthonormal System of the Hilbert Space H

Conclusions and Discussions