Pointwise Remez- and Nikolskii-type inequalities for exponential sums
Let \(E_n= \left\{f: f(t)= a_0+ \sum^n_{j=1} a_j e^{\lambda_jt}, a_j, \lambda_j\in\mathbb{R}\right\}\). In this paper the authors prove the following two inequalities, the first being of Remez-type while the second one is of Nikolskii-type. Theorem 1. Let \(s\in\left(0,{1\over 2}\right]\). There are absolute constants \(c_1>0\) and \(c_2> 0\) such that \[ \exp(c_1 ns)\leq \sup_f|f(0)|\leq \exp(c_2ns), \] where the supremum is taken for all \(f\in E_n\) satisfying \[ m(\{x\in [-1,1]: |f(x)|\leq 1\})\geq 2-s. \] Theorem 2. There are absolute constants \(c_1> 0\) and \(c_2> 0\) such that \[ c^{1+1/q}_1 \Biggl({1+ qn\over \min\{y- a,b-y\}}\Biggr)^{1/q}\leq \sup_{0\neq f\in E_n} {|f(y)|\over\|f\|_{L_q[a,b]}}\leq \Biggl({c_2(1+ qn)\over \min\{y-a, b-y\}}\Biggr)^{1/q} \] for every \(a< y< b\) and \(q>0\).
scientific article; zbMATH DE number 1406570