Embedding of weak Markov systems
Let A be a subset of the real line having at least \(n+2\) elements (\(n\geq 0)\), let I be the convex hull of A and let \(Z_ n=\{z_ 0,...,z_ n\}\) be a sequence of linearly independent real valued functions defined on A. If \(\{z_ 0,...,z_ i\}\) is a weak Chebyshev system for \(i=0,...,n\), then \(Z_ n\) is called a weak Markov system. A normed weak Markov system is a weak Markov system \(Z_ n\) for which \(z_ 0\equiv 1\). If every element of \(Z_ n\) is bounded in the intersection of A with any compact subset of I, we say that \(Z_ n\) is C-bounded on A. If \(U_ n=\{u_ 0,...,u_ n\}\) is a set of real valued functions defined on a real set A and \(V_ n=\{v_ 0,...,v_ n\}\) is a set of real valued functions defined on a real set B we say that \(U_ n\) can be embedded in \(V_ n\) if there is a strictly increasing function h:\(A\to B\) such that \(v_ i[h(t)]=u_ i(t)\) for every t in A and \(i=0,...,n\). The function h is called an embedding function. We have: Theorem: A normed weak Markov system \(U_ n\) on a set A can be embedded in a normed weak Markov system of continuous functions defined on an open bounded interval if and only if \(U_ n\) is C-bounded on A. Moreover if c is an arbitrary element of A, the embedding function h can be chosen so that \(h(c)=c\). The following correction should be brought to the readers' attention: On p. 253, lines 13 and 14, the sentence that begins with: ''Define \(y_ i\) on \(A^ c...''\) should be changed to the following: ''Define \(y_ i\) on \(A^ c\) as follows: If t is in A, \(y_ i(x)=u_ 1(x)\); on the other hand, if t is a point of accumulation of A that does not belong to A \(y_ i(x)=\lim_{k\to \infty}u_ i(x_ k)\) where \(\{x_ k\}\) is a sequence of points of A that converges to t and such that \(\{u_ i(x_ k)\}\) is convergent for all i, 0\(\leq i\leq n.''\) The reason for this change is that if \(y_ i(x)\) is defined as lim \(\sup_{x\to t}u_ i(x)\) as was done originally, then all we can say is that for each i there is a sequence \(\{x_ k^{(i)}\}\) such that \(y_ i(x)=\lim u_ i(x_ k^{(i)}).\) Since the sequences \(\{x_ k^{(i)}\}\) might be all different, this is not enough to guarantee that \(\{y_ 0,...,y_ n\}\) is a weak Markov system.
scientific article; zbMATH DE number 3857685