On uniformly continuous almost best generalized rational approximation operators (Q1957028)

For finite-dimensional subspaces \(V\) and \(W\) of \(C(K)\) -- for a compact metric space \(K\) -- let \(R(V/W)\) be the closure in \(C(K)\) of \[ \{v/w : v\in V,\;w\in W \text{ and } w > 0 \text{ on } K\}. \] This work identifies the conditions under which, for \(\varepsilon > 0\), there is a continuous (or uniformly continuous) mapping \(P(f)\) of \(C(K)\) into \(R(V/W)\) such that \[ \| f- P(f)\| < (1+\varepsilon) \text{dist}(f, R(V/W)). \]