Biased varisolvent Chebyshev approximation on subsets (Q1120087)

For a space of parameters P let \(F=\{F(A,\cdot):A\in P\}\) be a family ``unisolvent of variable degree'' of functions in C[\(\alpha\),\(\beta\) ] and let \(f\in C[\alpha,\beta]\) be given. The main result of the paper deals with the best one-sided approximation and another kind of approximation (in ``r-biased'' Chebyshev norm) of f by elements in F. Let \(\{X_ k\}_{k\geq 1}\) be a sequence of subsets of [\(\alpha\),\(\beta\) ] and \(\{r_ k\}_{k\geq 1}\) be a sequence of positive numbers. Denote by \(F(A_ k,\cdot)\) the best approximation of f on \(X_ k\) in the \(r_ k\)-biased norm, \(k\geq 1\) and by F(A,\(\cdot)\) the best one-sided approximation of f on [\(\alpha\),\(\beta\) ]. If \(\{X_ k\}\to [\alpha,\beta]\) then under some conditions the authors obtain the uniform convergence on [\(\alpha\),\(\beta\) ] of \(\{F(A_ k,\cdot)\}\) to F(A,\(\cdot)\).