Embedding a function into a Haar space (Q1114104)

The author proves that a continuous function f: \(R\to R\), M an interval in R, can be embedded into a Markov system \(\{f_ 1,...,f_ n\}\) of continuous functions on M, with \(f_ 1\equiv 1\), if and only if f has no weak oscillation of length \(n+1\). A weak oscillation of length \(n+1\) of f is a sequence \(x_ 1<...<x_{n+1}\) in M such that the quantities \((- 1)^ i\) \(f(x_ i)-f(x_{i-1})\) are all nonnegative (or all nonpositive), \(i=2,...,n+1\). A necessary and sufficient condition for embedding f into a Chebyshev system on M is derived. As was shown by \textit{R. Haverkamp} and \textit{R. Zielke} [J. Approximation Theory 30, 155-156 (1980; Zbl 0453.41023)] the function \(g(t)=t^ 3\cdot [1+(t/2)+\cos (\pi /t)]\), \(t>0\), \(g(0)=0\), cannot be embedded into a Chebyshev system on [0,\(\infty)\).