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The abc-Problem for Gabor Systems and Uniform Sampling in Shift-Invariant Spaces

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Excursions in Harmonic Analysis, Volume 3

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

In this chapter, we identify ideal window functions χ I on finite intervals I and time–frequency shift lattices \(a {\mathbb{Z}}\times b{\mathbb{Z}}\) such that the corresponding Gabor systems

$$\mathcal G(\chi_I, a {\mathbb{Z}}\times b{\mathbb{Z}}):=\{e^{-2\pi i n bt} \chi_I(t- m a):\ (m, n)\in{\mathbb{Z}}\times{\mathbb{Z}}\}$$

are frames for \(L^2({\mathbb{R}})\). Also we consider a stable recovery of rectangular signals f in a shift-invariant space

$$V_2(\chi_I, b{\mathbb{Z}}):=\Big\{\sum_{\lambda\in b{\mathbb{Z}}} d(\lambda) \chi_I(t-\lambda): \sum_{\lambda\in b{\mathbb{Z}}} |d(\lambda)|^2<\infty\Big\}$$

from their equally-spaced samples \(f(t_0+\mu), \mu\in a{\mathbb{Z}}\), for arbitrary initial sampling position t 0.

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Author information

Authors and Affiliations

  1. School of Mathematical and Computational Science, Sun Yat-Sen University, 510275, Guangzhou, P. R. China

    Xin-Rong Dai

  2. Department of Mathematics, University of Central Florida, 32816, Orlando, FL, USA

    Qiyu Sun

Authors
  1. Xin-Rong Dai
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  2. Qiyu Sun
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Corresponding author

Correspondence to Xin-Rong Dai .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, Maryland, USA

    Radu Balan

  2. Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, Maryland, USA

    Matthew J. Begué

  3. Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, University of Maryland, Norbert Weiner Center, College Park, Maryland, USA

    Wojciech Czaja

  5. Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, Maryland, USA

    Kasso A. Okoudjou

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Dai, XR., Sun, Q. (2015). The abc-Problem for Gabor Systems and Uniform Sampling in Shift-Invariant Spaces. In: Balan, R., Begué, M., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 3. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-13230-3_8

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