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Acta Applicandae Mathematicae

, Volume 164, Issue 1, pp 65–81 | Cite as

Undersampled Windowed Exponentials and Their Applications

  • Chun-Kit Lai
  • Sui TangEmail author
Article
  • 72 Downloads

Abstract

We characterize the completeness and frame/basis property of a union of under-sampled windowed exponentials of the form
$$ {\mathcal{F}}(g): =\bigl\{ e^{2\pi i n x}: n\ge 0\bigr\} \cup \bigl\{ g(x)e^{2\pi i nx}: n< 0\bigr\} $$
for \(L^{2}[-1/2,1/2]\) by the spectra of the Toeplitz operators with the symbol \(g\). Using this characterization, we classify all real-valued functions \(g\) such that \({\mathcal{F}}(g)\) is complete or forms a frame/basis. Conversely, we use the classical non-harmonic Fourier series theory to determine all \(\xi \) such that the Toeplitz operators with the symbol \(e^{2\pi i \xi x}\) is injective or invertible. These results demonstrate an elegant interaction between frame theory of windowed exponentials and Toeplitz operators. Finally, we use our results to answer some open questions in dynamical sampling, and derivative samplings on Paley-Wiener spaces of bandlimited functions.

Keywords

Completeness Frames Spectra Toeplitz operators Windowed exponentials 

Mathematics Subject Classification

94O20 42C15 42C30 

Notes

Acknowledgement

We would like to thank anonymous reviewers for their very helpful comments. Sui Tang is supported by the AMS Simons travel grant.

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
  2. 2.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA

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