On the spectral synthesis property and its application to partial differential equations (Q1197461)

For a closed subset \(M\subset\mathbb{R}^ n\) denote \[ \begin{aligned} I(M) &= \{f|\;\widehat{f}\in L^ 1(\mathbb{R}^ n),\;f\mid M=0\},\\ J^ k(M) &= \{f|\;f\in C_ 0^ k(\mathbb{R}^ n),\;f\mid M=0\}.\end{aligned} \] The main result of the paper is Theorem. Let \(k\geq n+2\). If \(M\) is a compact \(C^ k\)-hypersurface in \(\mathbb{R}^ n\) with non-vanishing Gaussian curvature, then \(\overline {J^ k(M)} =I(M)\) where the closure is taken in the Fourier transformed \(L^ 1\)-norm. As an application, a uniqueness property for partial differential equations with constant coefficients is proved.