Best piecewise monotone uniform approximation (Q808361)

Let \(I=[a,b]\) and B(I) (C(I)) be the Banach space of all bounded (resp. continuous) real valued functions on I with the uniform norm. For \(n\geq 1\) let \(\Omega =\{p=(p_ 1,p_ 1,...,p_ n)\in {\mathbb{R}}^ n: a=p_ 0\leq p_ 1\leq...\leq p_ n=b\}\) be the set of all knot vectors. For a given \(p\in \Omega\) let \(I_ j=[p_{j-1},p_ j)\), \(j=1,...,n-1\) and \(I_ n=[p_{n-1},p_ n]\) and let \(K(p)=\{h\in B(I): (-1)^ jh\) is nonincreasing on \(I_ j\), \(1\leq j\leq n\}\). Let also \(K=\cup \{K(p):\) \(p\in \Omega \}\). The object of this paper is the study of best approximation of a function \(f\in C=C(I)\) by elements in K(p), K, K(p)\(\cap C\) or in \(K\cap C\). The author proves theorems concerning existence, characterization and nonuniqueness of the element of best approximation.