Upper bounds for the derivative of exponential sums

Abstract:

The equality \[ \sup \limits _p \frac {{|p’(a)|}}{{{{\left \| p \right \|}_{[a,b]}}}} = \frac {{2{n^2}}}{{b - a}}\] is shown, where the supremum is taken for all exponential sums p of the form \[ p(t) = {a_0} + \sum \limits _{j = 1}^n {{a_j}{e^{{\lambda _j}t}},\quad {a_j} \in {\mathbf {R}},} \] with nonnegative exponents ${\lambda _j}$. The inequalities \[ {\left \| {p’} \right \|_{[a + \delta ,b - \delta ]}} \leq 4{(n + 2)^3}{\delta ^{ - 1}}{\left \| p \right \|_{[a,b]}}\] and \[ {\left \| {p’} \right \|_{[a + \delta ,b - \delta ]}} \leq 4\sqrt 2 {(n + 2)^3}{\delta ^{ - 3/2}}{\left \| p \right \|_{{L_2}[a,b]}}\] are also proved for all exponential sums of the above form with arbitrary real exponents. These results improve inequalities of Lorentz and Schmidt and partially answer a question of Lorentz.