Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions

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• 作者：Markus Faulhuber^a ; ^{markus.faulhuber@univie.ac.at" class="auth_mail" title="E-mail the corresponding author} ; Stefan Steinerberger^b ; ^{stefan.steinerberger@yale.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Gabor frame ; Frame bounds ; Jacobi theta functions ; Log-convexity ; Log-concavity
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：407-422
• 全文大小：387 K

文摘

We study sharp frame bounds of Gabor frames for integer redundancy with the standard Gaussian window and prove that the square lattice optimizes both the lower and the upper frame bound among all rectangular lattices. This proves a conjecture of Floch, Alard & Berrou (as reformulated by Strohmer & Beaver). The proof is based on refined log-convexity/concavity estimates for the Jacobi theta functions class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303985&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303985&_rdoc=1&_issn=0022247X&md5=dbc3f0473361a90ec1b9f7d24977476b" title="Click to view the MathML source">θ₃class="mathContainer hidden">class="mathCode">${\theta }_3$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303985&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303985&_rdoc=1&_issn=0022247X&md5=9c1fa54ca8b0ab0b6baf28c9807fcd16" title="Click to view the MathML source">θ₄class="mathContainer hidden">class="mathCode">${\theta }_4$.