Integral representation and embedding of weak Markov systems
\textit{R. Zielke} [ibid. 44, 30-42 (1985; Zbl 0584.41013)] proved that a nondegenerate normalized weak Markov system is representable. In the present paper the author proves that the same property is valid for weakly nondegenerate normalized weak Markov systems. A system \(Z_ n=\{z_ 0,...,z_ n\}\) of real functions defined on a set \(A\subset {\mathbb{R}}\) is called weakly nondegenerate provided that the following condition is satisfied: If \(Z_ n\) is linearly independent on \([c<\infty)\cap A\), where c is an arbitrary point in the convex hull of A, then there exists a set \(U_ n\), obtained from \(Z_ n\) by a triangular linear transformation, such that for any sequence \(\{\) k(r); \(r=0,...,m\}\) with k(0)\(\geq 0\) and k(m)\(\leq n\) that is either strictly increasing or constant, the set \(\{u_{k(r)}\); \(r=0,...,m\}\) is a weak Markov system on [c,\(\infty)\cap A\); a similar condition is supposed if \(Z_ n\) is linearly independent on (-\(\infty,c]\cap A\). (Naturally one suppose that \(Z_ n\) is linearly independent on at least one of the sets [c,\(\infty)\cap A\) and (-\(\infty,c]\cap A\) for every real number c. Markov system \(=\) complete Chebyshev system.)
weakly nondegenerate normalized weak Markov systems
scientific article; zbMATH DE number 4127830