On duality in rational approximation (Q805952)

Let G and H be subspaces of C(K) for a compact set K. Let \(R(G,H)=\{\frac{g}{h}:\;g\in G,\quad h\in H,\quad h>0\}.\) If \(f\in C(K)\), a formula is given for the distance from f to R(G,H). This shows, for example, which functions are in R(G,H). The results are specialized to G and H being spaces of polynomials (with gaps) to generalize known Müntz type results by \textit{J. Bak} and \textit{D. J. Newman} [J. Approximation Theory 23, 155-157 (1978; Zbl 0385.41007)], and \textit{G. Somorjai} [Acta Math. Acad. Hungar 27, 197-199 (1976; Zbl 0333.41012)].