Upper bounds for the derivative of exponential sums

Authors:
Peter Borwein and Tamás Erdélyi

Journal:
Proc. Amer. Math. Soc. **123** (1995), 1481-1486

MSC:
Primary 41A17; Secondary 41A10

DOI:
https://doi.org/10.1090/S0002-9939-1995-1232137-X

MathSciNet review:
1232137

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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.

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Article copyright:
© Copyright 1995
American Mathematical Society