On rational lacunary approximation on the interval \([-1, 1]\) (Q1890549)

The author obtain some Müntz-like theorems characterizing the approximation of continuous functions on \([-1,1]\) by rational combinations of \(\{x^{n_ j}\}^ \infty_{j = 1}\) for a given increasing sequence of integers \(\{n_ j\}\) (i.e. so-called rational lacunary combination in \(C_{[-1, 1]})\). The sufficient conditions relating odd and even numbers \(n_ j\) are given to have a dense set of the above combinations in \(C_{[-1, 1]}\) or not dense.