## Rocky Mountain Journal of Mathematics

### On Schauder basis properties of multiply generated Gabor systems

Morten Nielsen

#### Abstract

Let $\mathcal{A}$ be a finite subset of $L^2(\mathbb{R} )$ and $p,q\in \mathbb{N}$. We characterize the Schauder basis properties in $L^2(\mathbb{R} )$ of the Gabor system $G(1,p/q,\mathcal{A} )=\{e^{2\pi i m x}g(x-np/q) : m,n\in \mathbb{Z} , g\in \mathcal{A} \},$ with a specific ordering on $\mathbb{Z} \times \mathbb{Z} \times \mathcal{A}$. The characterization is given in terms of a Muckenhoupt matrix $A_2$ condition on an associated Zibulski-Zeevi type matrix.

#### Article information

Source
Rocky Mountain J. Math., Volume 46, Number 6 (2016), 2043-2060.

Dates
First available in Project Euclid: 4 January 2017

https://projecteuclid.org/euclid.rmjm/1483520437

Digital Object Identifier
doi:10.1216/RMJ-2016-46-6-2043

Mathematical Reviews number (MathSciNet)
MR3591271

Zentralblatt MATH identifier
1357.42034

#### Citation

Nielsen, Morten. On Schauder basis properties of multiply generated Gabor systems. Rocky Mountain J. Math. 46 (2016), no. 6, 2043--2060. doi:10.1216/RMJ-2016-46-6-2043. https://projecteuclid.org/euclid.rmjm/1483520437

#### References

• M. Bownik and O. Christensen. Characterization and perturbation of Gabor frame sequences with rational parameters, J. Approx. Th. 147 (2007), 67–80.
• O. Christensen, Frames and bases, in Applied and numerical harmonic analysis, An introductory course, Birkhäuser Boston, Inc., Boston, 2008.
• I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271–1283.
• H.G. Feichtinger and T. Strohmer, eds., Advances in Gabor analysis, in Applied and numerical harmonic analysis, Birkhäuser Boston, Inc., Boston, 2003.
• J.-P. Gabardo and D. Han, Balian-Low phenomenon for subspace Gabor frames, J. Math. Phys. 45 (2004), 3362–3378.
• C. Heil, A basis theory primer, in Applied and numerical harmonic analysis, expanded edition, Birkhäuser/Springer, New York, 2011.
• C. Heil and A.M. Powell, Gabor Schauder bases and the Balian-Low theorem, J. Math. Phys. 47 (2006), article id no. 113506.
• R. Hunt, B. Muckenhoupt and R. Wheeden. Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251.
• M. Nielsen, On stability of finitely generated shift-invariant systems, J. Fourier Anal. Appl. 16 (2010), 901–920.
• A. Ron and Z. Shen, Weyl-Heisenberg frames and Riesz bases in $L_2(\mathbf{R^d})$, Duke Math. J. 89 (1997), 237–282.
• E.M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. 43, Princeton University Press, Princeton, 1993.
• S. Treil and A. Volberg, Continuous frame decomposition and a vector Hunt-Muckenhoupt-Wheeden theorem, Ark. Mat. 35 (1997), 363–386.
• ––––, Wavelets and the angle between past and future, J. Funct. Anal. 143 (1997), 269–308.
• M. Zibulski and Y.Y. Zeevi, Analysis of multiwindow Gabor-type schemes by frame methods, Appl. Comp. Harmonic Anal. 4 (1997), 188–221.