Bandlimited approximations to the truncated Gaussian and applications

Abstract

In this paper we extend the theory of optimal approximations of functions $f:\mathbb{R}\to \mathbb{R}$ in the $L^1(\mathbb{R})$-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.

to appear in Const. Approx

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Cited by in corpus (11)

• Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function
• A note on the zeros of zeta and $L$-functions
• Extremal functions in de Branges and Euclidean spaces
• Bandlimited approximations and estimates for the Riemann zeta-function
• On the argument of $L$-functions
• Bounding $S_n(t)$ on the Riemann hypothesis
• Extremal problems in de Branges spaces: the case of truncated and odd functions
• Extremal functions in de Branges and Euclidean spaces II
• Entire approximations for a class of truncated and odd functions
• A Variation on Selberg's Approximation Problem
• One-sided Band-limited Approximations of Some Radial Functions