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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

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


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

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.

References [Enhancements On Off] (What's this?)

  • [1] S. Karlin and W. J. Studden, Tchebycheff systems: with applications in analysis and statistics, Interscience, New York, 1966. MR 0204922 (34:4757)
  • [2] G. G. Lorentz, Notes on approximation, J. Approx. Theory 56 (1989), 360-365. MR 990350 (90f:41004)
  • [3] E. Schmidt, Zur Kompaktheit der Exponentialsummen, J. Approx. Theory 3 (1970), 445-459. MR 0271588 (42:6471)
  • [4] P. W. Smith, An improvement theorem for Descartes systems, Proc. Amer. Math. Soc. 70 (1978), 26-30. MR 0467118 (57:6985)

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