Topology of the space of locally separable functions
The set \(\mathcal S\) of real \(C^\infty\)-functions of \(n\) variables which are of the form \(H(x)=u_1(x)+\dots+u_n(x)\) on some neighbourhood \(V\subset \mathbb R^n\) of the origin for some diffeomorphism \(u:V \to W\) such that \(u(0)=0\), is investigated. It is shown that \(\mathcal S\) contains an open dense subset of the Frechet space \(C^\infty(U,\mathbb R)\) for any fixed open neighbourhood \(U\) of the origin. The interior of \(\mathcal S\) is studied in more detail. Using the classification of the functions by the properties of the \(k\)-homogeneous polynomials \(T_k (H)\) of the Taylor expansion of \(H\) at the origin, two main results on \(\mathcal S\) are deduced. For \(n=2\), the set \(\text{Int} (\mathcal S)\) is fully described as the union of open sets of functions -- \(O_1\) of those with nonzero \(T_1(H)\), \(O_2\) of those with nonzero \(T_2(H)\), and a particular subset of analalogically defined \(O_3\). For \(n\ge 3\), just two inclusions approximating \(\text{Int} (\mathcal S)\) are proved. Namely, \(O_1\cup O_2(n)\cup O_2(n-1)\subset \text{Int} (\mathcal S) \subset O_1\cup O_2(n)\cup \dots \cup O_2(r_n)\) for uniquely defined integer \(r_n<n\). Here, \(O_2(r)\) is the subset of elements of \(O_2\) for which the rank of the Hessian at the origin is \(r\).
\(C^\infty\)-functions of \(n\) variables
scientific article; zbMATH DE number 7375602