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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Upper bounds for the derivative of exponential sums
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by Peter Borwein and Tamás Erdélyi PDF
Proc. Amer. Math. Soc. 123 (1995), 1481-1486 Request permission

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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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