Interpolation by generalized polynomials with restricted ranges (Q1308794)

\textit{J. Briggs} and \textit{L. A. Rubel} [ibid. 30, 160-168 (1980; Zbl 0492.41006)] proved the existence of a non-negative polynomial of degree \(\leq n\) that interpolates a non-negative continuous function at \(n+1\) distinct points. A similar result was found by \textit{A. Howitz} [ibid. 62, No. 1, 39-46 (1990; Zbl 0737.41003)] for interpolation by polynomials with restricted ranges. He asked some related questions. In this paper, by using some perturbation techniques, different from Hurwitz', the author gives affirmative answers to those questions. In fact he proves the following Theorem: Let \(\{\varphi_ 0,\dots, \varphi_ n\}\) be an extended Chebyshev system of order two on \([a,b]\) and \(p_ 1,p_{-1}\in \Phi_ n= \text{span} \{\varphi_ 0,\dots, \varphi_ n\}\) be subject to the condition \(p_ 1(x)< p_{-1}(x)\) on \([a,b]\). If \(f\in C[a,b]\) satisfies on this interval the condition \(p_ 1(x)\leq f(x)\leq p_{-1}(x)\), then there exists a polynomial \(p\in K(\ell,u)\), where \(K(\ell,u)= \{p\in \Phi_ n\), \(\ell(x)\leq p(x)\leq u(x)\), \(x\in [a,b]\}\), which interpolates \(f\) at \(n+1\) distinct points from \([a,b]\).