Bandlimited approximations to the truncated Gaussian and applications

In this paper we extend the theory of optimal approximations of functions $f: \R \to \R$ in the $L^1(\R)$-metric by entire functions of prescribed exponential type (bandlimited functions). We solve this problem for the truncated and the odd Gaussians using explicit integral representations and fine properties of truncated theta functions obtained via the maximum principle for the heat operator. As applications, we recover most of the previously known examples in the literature and further extend the class of truncated and odd functions for which this extremal problem can be solved, by integration on the free parameter and the use of tempered distribution arguments. This is the counterpart of the work \cite{CLV}, where the case of even functions is treated.Comment: to appear in Const. Appro

SHIFTED MOMENTS OF ‐FUNCTIONS AND MOMENTS OF THETA FUNCTIONS

Assuming the Riemann Hypothesis, Soundararajan [Ann. of Math.a (2) 170 (2009), 981-993] showed that fT0 ℓ.1=2 C it/2κ T .log T /κk2+€ His method was used by Chandee [Q.A J. Math. 62 (2011), 545-572] to obtain upper bounds for shifted moments of the Riemann Zeta function. Building on these ideas of Chandee and Soundararajan, we obtain, conditionally, upper bounds for shifted moments of Dirichlet-functions which allow us to derive upper bounds for moments of theta functions