On function spaces that are interpolating at any \(k\) nodes (Q1324676)

The following problem is considered: to define the minimal dimension \(I(M,k)\) of a linear subspace \(L\) in the space of continuous real-valued functions on a topological space \(M\) such that any function \(M\to\mathbb{R}\) may be interpolated at any distinct points of \(M\) by an appropriate function from \(L\). Main results: \(2k- d(k)\leq I(\mathbb{R}^ 2,k)\leq 2k-1\), where \(d(k)\) is the number of ones in the binary representation of \(k\); for any \(n\)-dimensional manifold \(M\leq I(M,k)\leq k(n+1)\). The methods of proofs use the multiplicative structure in the cohomology of configuration spaces \(B(M,k)\), i.e. spaces of all subsets of cardinality \(k\) in \(M\), and the Stiefel-Whitney classes of the bundle \(T(M,k)\) over \(B(M,k)\) whose fiber over the point \(\{x_ 1,\dots,x_ k\}\) is the space of real-valued functions on the set \(\{x_ 1,\dots,x_ k\}\).