Let \(\{f_1 (q), \dots, f_n(q)\}\) be a linearly independent system of continuous functions on any compact set \(Q\). The author introduces the notion of locally analytic functions and considers a ``polynomial'' \(P_\alpha (z)= \alpha_1 f_1 (z)+\dots +\alpha_n f_n (z)\), \(\alpha= (\alpha_1, \dots, \alpha_n)\in \mathbb{C}^n\). He describes the set \(Q\) where the following problem takes place: for any continuous function \(f(z)\) from \(C(Q)\) the unique ``polynomial'' \(P_\alpha (z)\) exists such that \[ \max_{z\in Q} |P_\alpha (z)- f(z)|- \min_\alpha\;\max_{z\in Q} |P_\alpha (z)- f(z)|. \]