Dependence of \(\alpha\) in peak norms and best peak norms approximation (Q1567429)

Let \(C([0,1])\) be the space of all continuous functions on \([0, 1]\). The paper is concerned with `peak norms: for \(0<\alpha\leq 1\) the \(\alpha\)-norm \(\|f\|_\alpha\) of \(f\) is defined to be \[ {1\over \alpha} \sup \Bigl \{\int_A|f|d\mu: A\subseteq [0,1],\mu(A)= \alpha\Bigr\} \] where \(\mu\) denotes Lebesgue measure. The \(\alpha\) norms were introduced in [\textit{E. Lapidot} and \textit{J. T. Levis} [J. Approximation Theory 67, 174-186 (1991; Zbl 0752.41026)] and were the subject of [C. Yang, Uniqueness of best \(\alpha\)-norm approximation and A-spaces, Numer. Funct. Anal. Optim. 13, 403-412 (1992; Zbl 0763.41027)]. Let \(U\) be a finite dimensional subspace of \(C([0,1])\). Then \(D_\alpha (f)=\inf \{\|f-u\|_\alpha: u\in U\}\) and \(P_\alpha (f)= \{p_\alpha \in U:\|f-p_\alpha \|_\alpha= D_\alpha(f)\}\). The paper considers \(\|f\|_\alpha\), \(D_\alpha (f)\) and \(P_\alpha (f)\) as functions of \(\alpha\), their continuity, differentiability and Lipschitz smoothness.