Limits of strong unicity constants for certain \(C^{\infty}\) functions (Q1058085)

Let C[-1,1] denote the space of all continuous real-valued functions on the interval [-1,1] endowed with the uniform norm, and let \(\Pi_ n\) be the subspace consisting of all polynomials of degree at most n. It is known that for each \(f\in C[-1,1]\) with best approximation \(B_ n(f)\) from \(\Pi_ n\) there is a smallest positive constant \(M_ n(f)\), called strong unicity constant, such that \(\| p-B_ n(f)\| \leq M_ n(f)[\| f-p\| -\| f-B_ n(f)\|]\) for all \(p\in \Pi_ n\). The authors' aim is to find classes of functions f for which \(\lim_{n\to \infty}M_ n(f)/n\) can be determined. Two such classes are pointed out. Firstly, it is proved that \(\lim_{n\to \infty}M_ n(f)/n=2(a+1)^{1/2}/(a-1)^{1/2}\) when f is the rational function defined by \(f(x)=1/(a-x)\) with \(a\geq 2\). Secondly, it is shown that for each f belonging to a certain class of continuously differentiable non- rational functions the equality \(\lim_{n\to \infty}M_ n(f)/n=2\) holds.