Markov and Bernstein type inequalities on subsets of \([-1,1]\) and \([- \pi,\pi]\)
The authors prove two theorems extending Markov's and Bernstein's inequalities to arbitrary subsets of \([-1,1]\) and \([-\pi,\pi]\), respectively. One of the theorems reads as follows: Let \(0< a\leq 2\pi\), \(0< L\leq 1\), let \(A\) be a closed subset of \([0,2\pi]\) with Lebesgue measure \(m(A)\geq 2\pi- a\). There is an absolute constant \(c> 0\) such that \[ \max_{t\in I} | p'(t)|\leq cL^{-1} (n+ n^ 2 a)\max_{t\in A} | p(t)| \] for every real trigonometric polynomial \(p\) of degree at most \(n\), and for every subinterval \(I\) of \(A\) with length at least \(La\).
scientific article; zbMATH DE number 638248