Bernstein type inequalities for a class of polynomials
Let n be a nonnegative integer, \(0<\omega \leq \pi\), and denote by \(T_ n\) the set of all real trigonometric polynomials of order at most n. Authors have introduced [Stud. Sci. Math. Hung. 24, No.1, 71-91 (1989; Zbl 0676.42003)] the following class of trigonometric polynomials: \[ {\mathcal T}_ n(\omega):=\{p(t)=\sum^{2n}_{j=0}a_ j \sin^ j\frac{\omega -t}{2}\sin^{2n-j}\frac{\omega +t}{2}| \text{ all }a_ j\geq 0\text{ or all }a_ j\leq 0\}. \] The aim of this paper is to prove Theorem 1. If \(q(t)=p(t)r(t)\), p(t)\(\in {\mathcal T}_ n(\omega)\), \(r(t)\in T_ k\) then \[ (*)\quad | q'(t)| \leq c_ 3(k+1)^ 2\sqrt{(n+1)/(\omega^ 2-t^ 2)}\| q\| \quad (| t| <\omega \leq \pi,\quad n,k\geq 0), \] where \(c_ 3\) is an absolute positive constant and \(\| \cdot \|\) denotes the sup norm over the inverval [-\(\omega\),\(\omega\) ]. Improvement \((k+1)^ 2\sqrt{n+1}\) to \(\sqrt{(n+k)(k+1)}\) in (*) is an open problem.
scientific article; zbMATH DE number 4208758