Uniform approximation on compact subsets of the real line

Let {? _i } _i=∩ ⁿ be continuous real functions on the compact set M?R. We consider the problem of best uniform approximation of the function? by polynomials \(\sum\nolimits_{i = 1}^n {c_i \varphi _i }\) on M. Let V(?₀, A) be a set of polynomials of best approximation on A ? M. We show that \(V(\varphi _0 ,M) = \mathop \cap \limits_{A_{n + 1} } V(\varphi _0 ,A_{n + 1} )\) , where A_n+1 represents all the possible sets of n+ 1 points {x₁, ..., x_n+1} in M, containing the characteristic set of the given problem of best approximation and for which the the rank of ∥?_i ∥ (i=1, ...,n; j=1,..., n+1) is equal to n. This theorem is applied to a problem of uniform approximation where {? _i } _i=1 ⁿ is a weakly Chebyshev system.