Summation of Gaussian shifts as Jacobi's third Theta function

Shengxin Zhu

doi:10.3934/mfc.2020015

Article Contents

2020, Volume 3, Issue 3: 157-163. Doi: 10.3934/mfc.2020015

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Summation of Gaussian shifts as Jacobi's third Theta function

Shengxin Zhu^,

Laboratory for Intelligent Computing and Financial Technology, Department of Mathematics, Xi'an Jiaotong-Liverpool University, Suzhou, 215123, China

^* Corresponding author: Shengxin Zhu
^* Corresponding author: Shengxin Zhu

Received: December 2019

Revised: May 2020

Published: August 2020

This research is supported by Foundation of LCP(6142A05180501), Jiangsu Science and Technology Basic Research Program (BK20171237), Key Program Special Fund of XJTLU (KSF-E-21, KSF-P-02), Research Development Fund of XJTLU (RDF-2017-02-23), and partially supported by NSFC (No.11771002, 11571047, 11671049, 11671051, 6162003, and 11871339)

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Abstract

A proper choice of parameters of the Jacobi modular identity (Jacobi Imaginary transformation) implies that the summation of Gaussian shifts on infinity periodic grids can be represented as the Jacobi's third Theta function. As such, connection between summation of Gaussian shifts and the solution to a Schrödinger equation is explicitly shown. A concise and controllable upper bound of the saturation error for approximating constant functions with summation of Gaussian shifts can be immediately obtained in terms of the underlying shape parameter of the Gaussian. This sheds light on how to choose a shape parameter and provides further understanding on using Gaussians with increasingly flatness.

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Mathematics Subject Classification: Primary: 65D05; Secondary: 65D05.

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Figure 1. $ \log_{10} \mathrm{csch} (\pi^2 d) $

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Figure 2. $ e^{-\frac{x^2}{d}} $

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Figure 3. error (-) and control line (-.-)

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