On characterization of best approximation with certain constraints (Q1266085)

Let \( \Phi_n := \text{span}\{\varphi_1,\dots,\varphi_n\}, \) be an \(n\)-dimensional subspace of real functions on \( [a,b] \) and assume that, for an integer \( k\geq 0, \) the \(k\)th derivatives \( \varphi_1^{(k)},\dots,\varphi_n^{(k)} \) are continuous. Denoting by \[ K_i=\{q\in \Phi_n: l_i(x)\leq q^{(i)}(x)\leq u_i(x), x\in [a,b]\}, \quad i=0,\dots,k, \] where \(l_i \) and \( u_i \) are extended real valued functions, let \( K_S=\cap_{i=0}^k K_i, \) and let \( K_\Lambda\subseteq \Phi_n \) be the so-called ``local convex cone'' at \( q_0\in K_S\). Under some conditions, the author obtains two characterization theorems for the best approximation element \( q_0\in K=K_S\cap K_\Lambda \) of the function \( f\in L_p[a,b]\setminus K \) respectively of \( f\in C(\mathcal X)\setminus K, \) where \(\mathcal X \) is a compact subset of \( [a,b] \) containing at least \( n+1 \) points. These general alternating theorems include as special cases, the approximation with interpolatory constraints, one-sided approximation and copositive approximation.