A note on Bernstein's theorems (Q1899345)

The authors repeat an argument of \textit{W. Pawłucki} and \textit{W. Pleśniak} [Math. Ann. 275, 467-480 (1986; Zbl 0591.32016)] and \textit{W. Pleśniak} [J. Approximation Theory 61, No. 1, 106-117 (1990; Zbl 0702.41023)] (case of \({\mathcal C}^\infty\) functions) to show that, for a compact subset \(E\) of \(\mathbb{R}^N\) preserving Markov's inequality with exponent \(r\), if the distance of \(f:\to \mathbb{R}\) from the polynomials on \(\mathbb{R}^N\) at degree at most \(n\) is of order \(n^{-r (k+n)}\) with some \(k\in \mathbb{Z}_+\) and \(\rho\in (0,1)\), then \(f\) extends to a function \(\widetilde {f}\) from a Zygmund-type class \(\Lambda_{k+ \rho} (\mathbb{R}^N)\). An analogical observation for Lipschitz-type classes has already been published by the second author [ibid. 67, No. 3, 252-269 (1991; Zbl 0752.41035)].