Optimal Tauberian constant in Ingham's theorem for Laplace transforms

Abstract

It is well known that there is an absolute constant $\mathfrak{C}>0$ such that if the Laplace transform $G(s)={\int }_0^{\infty }\rho (x)e^{-sx}\mathrm{d}x$ of a bounded function $\rho$ has analytic continuation through every point of the segment $(-i\lambda ,i\lambda )$ of the imaginary axis, then $$\underset{x\to \infty }{\mathrm{limsup}}\left|{\int }_0^x\rho (u)\mathrm{d}u-G(0)\right|\le \frac{\mathfrak{C}}{\lambda }\underset{x\to \infty }{\mathrm{limsup}}|\rho (x)|.$$ The best known value of the constant $\mathfrak{C}$ was so far $\mathfrak{C}=2$. In this article we show that the inequality holds with $\mathfrak{C}=\pi /2$ and that this value is best possible. We also sharpen Tauberian constants in finite forms of other related complex Tauberian theorems for Laplace transforms.

22 pages

References in corpus (1)

• Quantified versions of Ingham's theorem

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