Let A be a Banach algebra of functions, \(A\subset C_ 0(R^ n)\). Suppose \(x\in R^ n\) and denote by \(J_ x\) the closed ideal of all functions \(f\in A\) for which there exists a neighborhood G of the point x such that \(f(y)=0\), \(y\in G\). To any closed ideal \(I\subset A\) corresponds a closed ideal \(I_ x=\overline{I+J_ x}\). We say that in the algebra A spectral synthesis of ideals take place if for any closed ideal \(I\subset A\), \(I=\cap_{x\in R^ n}I_ x\). Whitney proved that spectral synthesis takes place in the algebra \(C_ 0^{m,\alpha}(R^ n)\). The author announces the following result: Let \(A=W_ p^{\ell}(R^ n)\) be Sobolev space, \(p\ell >n\) or \(p\ell =n\), \(p=1\). In this algebra the spectral synthesis of ideals take place iff \(n=1\), \(1\leq p<+\infty\) or \(n\geq 2\), \(n<p<+\infty\).