Yao Ihébami Akakpo, Mahouton Norbert Hounkonnou, Koffi Enakoutsa, Kofi V. S. Assiamoua
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引用次数: 0
摘要
在我们的研究中,我们拓宽了傅里叶-斯蒂尔杰斯变换的范围,将局部紧凑群(表示为 G)也包括在内。我们通过利用封闭子群 K 的诱导表示来实现这一扩展,并由此推导出定义在 G 上的哈尔积分函数 f 的傅里叶变换 ({\hat{f}}/)。具体来说,我们将 \({\hat{f}} 表示为度量 \(\mu = f \lambda \)的傅里叶-斯蒂尔杰斯变换 \({\hat{f}}\),其中 \( \lambda \)表示 G 的哈尔度量。我们的工作意义重大,因为当应用于具有紧凑子群 K 的李群时,我们的傅里叶-斯蒂尔杰斯变换 \({\hat{m}}\)与传统上通过 Gel'fand 变换定义的傅里叶-斯蒂尔杰斯变换相比,表现出更细微的特征,而后者在李群中是标准的。我们严格证实了这一观点。我们面临的主要挑战之一是构建 "三角函数",它是建立傅立叶变换的基础。
Development of a Fourier–Stieltjes transform using an induced representation on locally compact groups
In our research, we broaden the scope of Fourier–Stieltjes transforms to encompass locally compact groups, denoted as G. We achieve this extension by leveraging the induced representation from a closed subgroup K. From this, we deduce the Fourier transform \({\hat{f}}\) of a Haar-integrable function f defined on G. Specifically, we express \({\hat{f}}\) as the Fourier–Stieltjes transform \({\hat{\mu }}\) of the measure \(\mu = f \lambda \), where \( \lambda \) denotes the Haar measure of G. Our work is significant because when applied to Lie groups with compact subgroups K, our Fourier–Stieltjes transform \({\hat{m}}\) exhibits more nuanced characteristics compared to the traditionally defined one via the Gel’fand transform, which is standard in the context of Lie groups. We rigorously substantiate this observation. One of the principal challenges we confront is the construction of the “trigonometric functions”, which serve as the foundation for building the Fourier transform.