Acta Applicandae Mathematicae

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

Undersampled Windowed Exponentials and Their Applications

  • Chun-Kit Lai
  • Sui TangEmail author


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.


Completeness Frames Spectra Toeplitz operators Windowed exponentials 

Mathematics Subject Classification

94O20 42C15 42C30 



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


  1. 1.
    Aceska, R., Kim, Y.H.: Scalability of frames generated by dynamical operators. arXiv:1608.05622 (2016)
  2. 2.
    Aceska, R., Tang, S.: Dynamical sampling in hybrid shift invariant spaces. In: Operator Methods in Wavelets, Tilings, and Frames, vol. 626, p. 149. Am. Math. Soc., New York (2014) zbMATHGoogle Scholar
  3. 3.
    Acosta-Reyes, E., Aldroubi, A., Krishtal, I.: On stability of sampling-reconstruction models. Adv. Comput. Math. 31, 5–34 (2009) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Adcock, B., Gataric, M., Hansen, A.C.: Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples. Appl. Comput. Harmon. Anal. 42(3), 508–535 (2015) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Adcock, B., Gataric, M., Hansen, A.C.: Density theorems for nonuniform sampling of bandlimited functions using derivatives or bunched measurements. J. Fourier Anal. Appl. 23, 1311–1347 (2017) MathSciNetzbMATHGoogle Scholar
  6. 6.
    Adcock, B., Gataric, M., Romero, J.L.: Computing reconstructions from nonuniform Fourier samples: universality of stability barriers and stable sampling rates. Appl. Comput. Harmon. Anal. (2017). CrossRefzbMATHGoogle Scholar
  7. 7.
    Aldroubi, A., Cabrelli, C., Cakmak, A.F., Molter, U., Petrosyan, A.: Iterative actions of normal operators. J. Funct. Anal. 272, 1121–1146 (2017) MathSciNetzbMATHGoogle Scholar
  8. 8.
    Aldroubi, A., Cabrelli, C., Molter, U., Tang, S.: Dynamical sampling. Appl. Comput. Harmon. Anal. 42, 378–401 (2017) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Aldroubi, A., Davis, J., Krishtal, I.: Dynamical sampling: time-space trade-off. Appl. Comput. Harmon. Anal. 34, 495–503 (2013) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Aldroubi, A., Davis, J., Krishtal, I.: Exact reconstruction of signals in evolutionary systems via spatiotemporal trade-off. J. Fourier Anal. Appl. 21, 11–31 (2015) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Aldroubi, A., Gröchenig, K.: Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43, 585–620 (2001) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Balan, R.: Stability theorems for Fourier frames and wavelet Riesz bases. J. Fourier Anal. Appl. 3, 499–504 (1997) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Balan, R., Casazza, P.G., Heil, C., Landau, Z.: Density, overcompleteness, and localization of frames, I: theory. J. Fourier Anal. Appl. 12, 105–143 (2006) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Benedetto, J.J.: Irregular sampling and frames. In: Wavelets: a Tutorial in Theory and Applications, vol. 2, pp. 445–507 (1992) Google Scholar
  15. 15.
    Böttcher, A., Silbermann, B.: Analysis of Toeplitz operators. Springer, New York (2013) zbMATHGoogle Scholar
  16. 16.
    Casazza, P.G., Christensen, O., Kalton, N.J.: Frames of translates. Collect. Math. 52, 35–54 (2001) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Casazza, P.G., Christensen, O., Li, S., Lindner, A.: Density results for frames of exponentials. In: Harmonic Analysis and Applications, pp. 359–369. Springer, New York (2006) Google Scholar
  18. 18.
    Christensen, O.: An Introduction to Frames and Riesz Bases, vol. 7. Springer, New York (2003) zbMATHGoogle Scholar
  19. 19.
    Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952) MathSciNetzbMATHGoogle Scholar
  20. 20.
    Fogel, L.: A note on the sampling theorem. IRE Trans. Inf. Theory 1, 47–48 (1955) Google Scholar
  21. 21.
    Gabardo, J.-P., Lai, C.-K.: Frames of multi-windowed exponentials on subsets of rd. Appl. Comput. Harmon. Anal. 36, 461–472 (2014) MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gröchenig, K.: Reconstruction algorithms in irregular sampling. Math. Comput. 59, 181–194 (1992) MathSciNetzbMATHGoogle Scholar
  23. 23.
    Gröchenig, K.: Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type. Math. Comput. 68, 749–765 (1999) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Heil, C.: History and evolution of the density theorem for Gabor frames. J. Fourier Anal. Appl. 13, 113–166 (2007) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Heil, C.: A Basis Theory Primer: Expanded Edition. Springer, New York (2010) Google Scholar
  26. 26.
    Heil, C., Kutyniok, G.: Density of frames and Schauder bases of windowed exponentials. Houst. J. Math. 34, 565–600 (2008) MathSciNetzbMATHGoogle Scholar
  27. 27.
    Heil, C., Yoon, G.: Duals of windowed exponential systems. Acta Appl. Math. 119, 97–112 (2012) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Jaffard, S., et al.: A density criterion for frames of complex exponentials. Mich. Math. J. 38, 339–348 (1991) MathSciNetzbMATHGoogle Scholar
  29. 29.
    Jagerman, D., Fogel, L.: Some general aspects of the sampling theorem. IRE Trans. Inf. Theory 2, 139–146 (1956) Google Scholar
  30. 30.
    Kozma, G., Nitzan, S.: Combining Riesz bases. Invent. Math. 199, 267–285 (2015) MathSciNetzbMATHGoogle Scholar
  31. 31.
    Lai, C.-K.: On Fourier frame of absolutely continuous measures. J. Funct. Anal. 261, 2877–2889 (2011) MathSciNetzbMATHGoogle Scholar
  32. 32.
    Levinson, N.: Gap and Density Theorems, vol. 26. Am. Math. Soc., New York (1940) zbMATHGoogle Scholar
  33. 33.
    Linden, D., Abramson, N.M.: A generalization of the sampling theorem. Inf. Control 3, 26–31 (1960) MathSciNetzbMATHGoogle Scholar
  34. 34.
    Martínez-Avendaño, R.A., Rosenthal, P.: An Introduction to Operators on the Hardy-Hilbert Space, vol. 237. Springer, New York (2007) zbMATHGoogle Scholar
  35. 35.
    Nitzan, S., Olevskii, A., Ulanovskii, A.: Exponential frames on unbounded sets. Proc. Am. Math. Soc. 144, 109–118 (2016) MathSciNetzbMATHGoogle Scholar
  36. 36.
    Ortega-Cerdà, J., Seip, K.: Fourier frames. Ann. Math. 155, 789–806 (2002) MathSciNetzbMATHGoogle Scholar
  37. 37.
    Papoulis, A.: Generalized sampling expansion. IEEE Trans. Biomed. Circuits Syst. 24, 652–654 (1977) MathSciNetzbMATHGoogle Scholar
  38. 38.
    Philipp, F.: Bessel orbits of normal operators. J. Math. Anal. Appl. 448, 767–785 (2017) MathSciNetzbMATHGoogle Scholar
  39. 39.
    Rawn, M.D.: A stable nonuniform sampling expansion involving derivatives. IEEE Trans. Inf. Theory 35, 1223–1227 (1989) MathSciNetzbMATHGoogle Scholar
  40. 40.
    Razafinjatovo, H.N.: Iterative reconstructions in irregular sampling with derivatives. J. Fourier Anal. Appl. 1, 281–295 (1994) MathSciNetzbMATHGoogle Scholar
  41. 41.
    Redheffer, R.M., Young, R.M.: Completeness and basis properties of complex exponentials. Trans. Am. Math. Soc. 277, 93–111 (1983) MathSciNetzbMATHGoogle Scholar
  42. 42.
    Seip, K.: On the connection between exponential bases and certain related sequences in l2 (\(- \pi , \pi \)). J. Funct. Anal. 130, 131–160 (1995) MathSciNetzbMATHGoogle Scholar
  43. 43.
    Zibulski, M., Segalescu, V., Cohen, N., Zeevi, Y.: Frame analysis of irregular periodic sampling of signals and their derivatives. J. Fourier Anal. Appl. 2, 453–471 (1995) MathSciNetzbMATHGoogle Scholar

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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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