On a conjecture concerning strong unicity constants
Let \(P_n\) be the set of algebraic polynomials of degree at most \(n\in\mathbb{N}_0\) and let \(f\) be a given real-valued function \(f\in C[-1, 1]\). \textit{D. J. Newman} and \textit{H. S. Shapiro} [Duke Math. J. 30, 673-682 (1963; Zbl 0116.04502)] proved the theorem: for each \(n\in\mathbb{N}_0\) there exists a constant \(C= C_n(f)\) such that \[ \|f-q\|\geq\|f- q^*_n\|+ C\|q- q^*_n\| \] for all \(q\in P_n\), where \(q^*\) is the best uniform approximation of \(f\) in the set \(P_n\). In the present paper, the author gives the following definition: for each \(n\in\mathbb{N}_0\) the largest constant \(C\) such that the above inequality holds is called the strong unicity constant denoted by \(\gamma_n(f)\). The author proves that if \(f\) is not a polynomial, then \(\limsup_{n\to\infty} M_n(f)=\infty\), where \(M_n(f)= 1/\gamma_n(f)\). This result was conjectured by \textit{M. S. Henry} and \textit{J. A. Roulier} [J. Approximation theory 22, 85-94 (1978; Zbl 0384.41012)].
scientific article; zbMATH DE number 1398583
scientific article