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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
MathSciNet review: 1232137
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Abstract: The equality

$\displaystyle \mathop {\sup }\limits_p \frac{{\vert p'(a)\vert}}{{{{\left\Vert p \right\Vert}_{[a,b]}}}} = \frac{{2{n^2}}}{{b - a}}$

is shown, where the supremum is taken for all exponential sums p of the form

$\displaystyle 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

$\displaystyle {\left\Vert {p'} \right\Vert _{[a + \delta ,b - \delta ]}} \leq 4{(n + 2)^3}{\delta ^{ - 1}}{\left\Vert p \right\Vert _{[a,b]}}$

and

$\displaystyle {\left\Vert {p'} \right\Vert _{[a + \delta ,b - \delta ]}} \leq 4\sqrt 2 {(n + 2)^3}{\delta ^{ - 3/2}}{\left\Vert p \right\Vert _{{L_2}[a,b]}}$

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

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1232137-X
PII: S 0002-9939(1995)1232137-X
Article copyright: © Copyright 1995 American Mathematical Society