Regulated extensions in rational Chebyshev approximation

AbstractLet V(N, μ) ⊂ C(X) be a set of rationals of the form BLμ with B, L ϵ C(X), L(x) > 0 ∀x ϵ X, and μ ϵ N; we study existence of best approximations for extensions of V(N, μ) into the space of regulated functions R(X). We show existence of best approximations for an extension V0(N, μ) which is maximal in the sense that the quality of approximation on V0(N, μ) is the best we can achieve by any proper rational extension of V(N, μ).Then we consider the problem of characterization of minimal extensions of V(N, μ) which possess stabilized best approximations. We show that a quasiminimal extension V0(N, μ) similar to that defined by Rice [4, p. 82] in general is no minimal extension and give a necessary and sufficient condition for V0(N, 1) being minimal.Finally we want to give some sufficient conditions under which V0(N, μ) is a minimal extension for large enough μ ϵ N