Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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A BANACH ALGEBRA AND ITS EQUIVALENT SPACES OVER PATHS WITH A POSITIVE MEASURE

  • Received : 2019.09.03
  • Accepted : 2019.12.12
  • Published : 2020.07.31
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Abstract

Let C[0, T] denote the space of continuous, real-valued functions on the interval [0, T] and let C0[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 𝓕(𝓗), where 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 C0[0, T], ${\bar{\mathcal{F}}}({\mathcal{H}}) $ and ${\bar{\mathcal{S}}}_{{\alpha},{\beta};{\varphi}}$ reduce to 𝓕(𝓗) and Cameron-Storvick's Banach algebra 𝓢, respectively, which is the space of generalized Fourier-Stieltjes transforms of the complex-valued, finite Borel measures on L2[0, T].

Keywords

Acknowledgement

This work was supported by Kyonggi University Research Grant 2018.

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