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 \({\hat{f}}\) $\widehat{f}$ of a Haar-integrable function f defined on G. Specifically, we express \({\hat{f}}\) $\widehat{f}$ as the Fourier–Stieltjes transform \({\hat{\mu }}\) $\widehat{\mu }$ of the measure \(\mu = f \lambda \) $\mu =f\lambda$ , where \( \lambda \) $\lambda$ denotes the Haar measure of G. Our work is significant because when applied to Lie groups with compact subgroups K, our Fourier–Stieltjes transform \({\hat{m}}\) $\widehat{m}$ 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.