2016/10/31 by Michael Speckbacher, Peter Balazs, Péter Balázs
Computer Science · Mathematics · #Computer science #Digital Filter Design and Implementation #Function (biology) #Gabor transform #Gaussian #Generalization #Geometry #Image and Signal Denoising Methods #Integer (computer science) #Integrable system #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematics #Pure mathematics #Square (algebra) #Square-integrable function #Time–frequency analysis #math.FA #msc:42C15 #msc:42C40
paper · pdf · doi:10.1016/j.jmaa.2017.05.079
published as Journal of Mathematical Analysis and Applications, Vo. 455(2), pp. 1072-1087 (2017)
arxiv created 2017/01/12 · openalex publication_date 2017/06/07 · arxiv updated 2019/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We use the concept of reproducing pairs to study Gabor systems at critical density. First, we present a generalization of the Balian-Low theorem to the reproducing pairs setting. Then, we prove our main result that there exists a reproducing partner for the Gabor system of integer time-frequency shifts of the Gaussian. In other words, the coefficients for this Gabor expansion of a square integrable function can be calculated using inner products with an unstructured family of vectors in L2(\R). This solves the possibly last open question for this system.