Entire majorants via Euler–Maclaurin summation

Littmann, Friedrich

doi:10.1090/S0002-9947-06-04121-3

Entire majorants via Euler–Maclaurin summation
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Trans. Amer. Math. Soc. 358 (2006), 2821-2836 Request permission

Abstract:

It is the aim of this article to give extremal majorants of type $2\pi \delta$ for the class of functions $f_n(x)=\text {sgn}(x)x^n$, where $n\in \mathbb {N}$. As applications we obtain positive definite extensions to $\mathbb {R}$ of $\pm (it)^{-m}$ defined on $\mathbb {R}\backslash [-1,1]$, where $m\in \mathbb {N}$, optimal bounds in Hilbert-type inequalities for the class of functions $(it)^{-m}$, and majorants of type $2\pi$ for functions whose graphs are trapezoids.

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