Best local approximation and generalized convexity (Q1099344)

\textit{C. K. Chui, O. Shisha} and \textit{P. W. Smith} in ibid. 15, 371-381 (1975; Zbl 0314.41016) proved that if \(\{u_ 0,u_ 1,...,u_{n-1}\}\) is a T-system on [a,b] with \(W(u_ 0,u_ 1,...,u_{n-1},a)=\det (u_ i^{(j)}(a))_{i,j=0},\) then the net \(\{T_{n-1}(f;[a,a+\epsilon])\}\) of best uniform approximations to \(f\in C^{n-1}[a,b]\) from \(\Lambda_{n-1}=span\{u_ 0,u_ 1,...,u_{n-1}\}\) converges as \(\epsilon\to 0\) \(+\). The limit function \(T\) \(a_{n-1}f\) is the element of \(\Lambda_{n-1}\) that satisfies \((T\quad a_{n- 1}f)^{(j)}(a)=f^{(j)}(a),\) \(j=0,1,...,n-1\). In this note is shown that the same result holds for the sign-monotone norms (a norm is said to be sign-monotone on C[a,b] if f(x)g(x)\(\geq 0\) and \(| f(x)| \geq | g(x)|\) for all \(x\in [a,b]\) imply \(\| f\| \geq \| g\|)\) on C[a,b]. The generalized convexity with respect to \(\{u_ 0,...,u_{n-1}\}\) of a function f, by its best local approximations is also characterized.