A Banach algebra and its equivalent spaces over paths with a positive measure

Commun. Korean Math. Soc. 2020 Vol. 35, No. 3, 809-823 https://doi.org/10.4134/CKMS.c190314 Published online March 10, 2020 Printed July 31, 2020

Dong Hyun Cho Kyonggi University

Abstract : Let $C[0,T]$ denote the space of continuous, real-valued functions on the interval $[0,T]$ and let $C_0[0,T]$ be the space of functions $x$ in $C[0,T]$ with $x(0)=0$. In this paper, we introduce a Banach algebra $\bar{\mathcal S}_{\alpha,\beta;\varphi}$ on $C[0,T]$ and its equivalent space $\bar{\mathcal F}(\mathcal H)$, a space of transforms of equivalence classes of measures, which generalizes Fresnel class ${\mathcal F}(\mathcal H)$, where $\mathcal H$ is an appropriate real separable Hilbert space of functions on $[0,T]$. We also investigate their properties and derive an isomorphism between $\bar{\mathcal S}_{\alpha,\beta;\varphi}$ and $\bar{\mathcal F}(\mathcal H)$. When $C[0,T]$ is replaced by $C_0[0,T]$, $\bar{\mathcal F}(\mathcal H)$ and $\bar{\mathcal S}_{\alpha,\beta;\varphi}$ reduce to ${\mathcal F}(\mathcal H)$ and Cameron-Storvick's Banach algebra $\mathcal S$, respectively, which is the space of generalized Fourier-Stieltjes transforms of the complex-valued, finite Borel measures on $L^2[0,T]$.

Keywords : Analytic Feynman integral, Banach algebra, Fresnel class, It\^o integral, Paley-Wiener-Zygmund integral, Wiener space