On extending the domain of definition of Chebyshev and weak Chebyshev systems (Q1825394)

The authors study the problem of extending the domain of definition of Haar or weak Haar spaces. Let M denote a set of real numbers having at least \(n+1\) poits and let S be an n-dimensional Haar (or Markov) space defined on M. We say that S can be continued to the left if there is an n-dimensional Haar (or Markov) space U defined on a set of the form (d,a]\(\cup M\), with \(d<a\), where \(a:=\inf (M)>-\infty\), such that the restriction of U to M coincides with S. If S is an n-dimensional subspace of the set of real functions on M we say that S is an E-space if it has a basis \(Z_ n=\{z_ 1,...,z_ n\}\) such that for any integers \(1\leq r(1)<...<r(m)\leq n\), \(\{z_{r(k)}:\) \(k=1,...,m\}\) is a Markov system. The authors prove the following theorem: Let S be an n-dimensional Haar space defined on a set M having the property that between any two elements of M there is a third element of M, and such that \(\inf (M)>- \infty\). Assume moreover that if an endpoint of M belongs to M, then it is a point of accumulation of M and all the functions in S are continuous there. Then S can be continued to the left to a Markov space if and only if S is an E-space. The authors include vivid historical remarks on the questions addressed in this paper and provide new examples illustrating the concepts discussed, such as an example of a continuous normalized Markov system on a closed interval that cannot be extrapolated to a larger interval as a Markov system, and an example of a Markov space on an open interval that contains no basis that is an E-system.