The density of rational functions in Markov systems: A counterexample to a conjecture of D. J. Newman (Q1205140)

The Muntz Theorem states that linear combinations of \(\{1,x^{u_ 1},x^{u_ 2},\dots\}\) are dense in \(C[0,1]\) if and only if \(\sum{1\over u_ k}<\infty\). However the family of quotients of such functions is always dense. That is, the rational functions \({\{a_ 0+\sum^ n a_ kx^{u_ k}\over b_ 0+\sum^ m b_ kx^{u_ k}\}}\) are dense in \(C[0,1]\) with no restriction on the distinct, real exponents \(u_ k\). This was conjectured by D. J. Newman and proved by \textit{J. Bak} and \textit{D. J. Newman} [J. Approximation Theory 23, 155-157 (1973; Zbl 0385.41007)] with a significant contribution by \textit{G. Somorjai} [Acta Math. Acad. Sci. Hungar. 27, 197-199 (1976; Zbl 0333.41012)]. In 1979 Newman asked what the limits for such generalizations might be. For example, he conjectured that perhaps the quotient of members from any Markov system might be dense. This paper nicely presents a counterexample to this last possibility. The construction is basic and involves functions built from a lacunary series of polynomials.