Acta Applicandae Mathematicae

, Volume 160, Issue 1, pp 53–65 | Cite as

Functional Matrix Multipliers for Parseval Gabor Multi-frame Generators

  • Zhongyan LiEmail author
  • Deguang Han


We consider the problem of characterizing the bounded linear operator multipliers on \(L^{2}(\mathbb{R})\) that map Gabor frame generators to Gabor frame generators. We prove that a functional matrix \(M(t)=[f_{ij}(t)]_{m \times m}\) (where \(f_{ij}\in L^{\infty}(\mathbb{R})\)) is a multiplier for Parseval Gabor multi-frame generators with parameters \(a, b >0\) if and only if \(M(t)\) is unitary and \(M^{*}(t)M(t+\frac{1}{b})= \lambda(t)I\) for some unimodular \(a\)-periodic function \(\lambda(t)\). As a special case (\(m =1\)) this recovers the characterization of functional multipliers for Parseval Gabor frames with single function generators.


Gabor family Gabor multi-frame generators Functional matrix multipliers 

Mathematics Subject Classification (2010)

42C15 46C05 47B10 



The authors would like to thank the referee for several helpful comments and suggestions that helped us improve the presentation of this paper.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsNorth China Electric Power UniversityBeijingChina
  2. 2.Department of MathematicsUniversity of Central FloridaOrlandoUSA

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