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

Abstract

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.