Gordon's Conjecture 3: Fourier transforms in the hyperfinite setting

Pavol Jan Zlatos


Using methods of nonstandard analysis and building upon the results of our previous paper in which we proved Gordon's Conjectures~1 and~2 we will show that for any locally compact abelian group $\mathbf G$ the Fourier transform $\mathbf F\colon \operatorname{L}^1(\mathbf G) \to \operatorname{C}_0\bigl(\widehat{\mathbf G}\bigr)$, the Fourier-Stieltjes transform $\mathbf F\colon \operatorname{M}(\mathbf G) \to \operatorname{C}_{\operatorname{bu}}\bigl(\widehat{\mathbf G}\bigr)$, as well as all the generalized Fourier transforms $\mathbf F\colon \operatorname{L}^p(\mathbf G) \to \operatorname{L}^q\bigl(\widehat{\mathbf G}\bigr)$ or any pair of adjoint exponents $p \in (1,2]$, $q \in [2, \infty$ can be approximated in a fairly natural way by the discrete Fourier transform $\mathcal F\colon \Bbb C^G \to \Bbb C^{\widehat{G\,}}$ on a (hyper)finite abelian group $G$. In particular, we will prove Gordon's Conjecture~3, originally stated for the Fourier-Plancherel transform $\mathbf F\colon \operatorname{L}^2(\mathbf G) \to \Lb^2\bigl(\widehat{\mathbf G}\bigr)$, and generalize it to all the above mentioned cases. Some standard consequences will be considered, as well.


Locally compact abelian group, Pontryagin–van Kampen duality, Fourier transform, nonstandard analysis, hyperfinite, infinitesimal, approximation

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DOI: https://doi.org/10.4115/jla.2021.13.7

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Journal of Logic and Analysis ISSN:  1759-9008