Notes on a classic theorem of Erdős and Grünwald (Q544025)

\textit{P. Erdős} and \textit{G. Grünwald} [Bull. Am. Math. Soc. 44, 515--518 (1938; Zbl 0019.11103)] proved what is known as the Erdős-Grünwald theorem \[ |\ell_{k,n}(x)|\leq {4\over \pi},\;|x|\leq 1,\;1\leq k\leq n,\;n=1,2,\ldots, \] where \[ \ell_{k,n}(x)={(-1)^{k+1}\cos{nt}\sin{t_{k,n}}\over n(\cos{t}-\cos{t_{k,n}})},\;x=\cos{t},\;k=1,\ldots n, \] with \[ x_{k,n}=\cos{t_{k,n}},\;t_{k,n}={2k-1\over 2n}\,\pi,\;k=1,\ldots,n. \] (i.e., the fundamental polynomials of Lagrange interpolation on the zeros of the Chebyshev polynomials) As a corollary it was proved \[ \lim_{n\rightarrow\infty}\max_{|x|\leq 1}\,\,\max_{1\leq k\leq n}\,\ell_{k,n}(x)={4\over\pi}=1.273\ldots \] In the paper under review (`a little gem') the author gives an elementary proof of the connected quantity where the maximum over the interval \([-1,1]\) is replaced by the minimum \[ \lim_{n\rightarrow\infty}\min_{|x|\leq 1}\,\,\max_{1\leq k\leq n}\,\ell_{k,n}(x)={2\over\pi}\,\cos{{2-\sqrt{3}\over 2}\pi} =0.580\ldots \]