A note on Lipschitz functions and spectral synthesis (Q579566)

A function \(\phi \in L^{\infty}({\mathbb{R}}^ n)\) is said to obey spectral synthesis, if \(f*\phi =0\) for each function \(f\in L^ 1({\mathbb{R}}^ n)\) whose Fourier transform \(\hat f\) vanishes on the spectrum \(\sigma(\phi)= supp{\hat \phi}\) of \(\phi\). The following results are shown: 1. There exists a function \(\phi\in \cap_{2<p\leq \infty}L^ p({\mathbb{R}}^ n)\) with compact spectrum which does not admit spectral synthesis. 2. If \(\phi \in L^ p\cap L^{\infty}({\mathbb{R}}^ n)\) for some \(2\leq p\leq \infty\), and if \(f\in L^ 1({\mathbb{R}}^ n)\) is such that \(\hat f(\sigma(\phi))=0\) and \(\hat f\in Lip_{n/2-n/p}({\mathbb{R}}^ n)\), then \(f*\phi =0.\) These results extend some similar results for the group of integers by \textit{J.-P. Kahane} and \textit{R. Salem} [Ensembles parfaits et séries trigonométriques (Hermann, Paris 1963; Zbl 0112.293)], and are derived by similar methods in combination with transference from \({\mathbb{T}}={\hat {\mathbb{Z}}}\) to \({\mathbb{R}}\).