Sampling and reconstruction in reproducing kernel subspaces of mixed Lebesgue spaces

Abstract

In this paper, we study the sampling and average sampling problems in a reproducing kernel subspace of mixed Lebesgue space. Let V be an image of \(L^{p,q}({\mathbb {R}}^{d+1})\) under idempotent integral operator defined by a kernel K satisfying certain decay and regularity conditions. Then, we prove that every f in V can be reconstructed uniquely and stably from its samples as well as from its average samples taken on a sufficiently small \(\gamma \)-dense set. Further, we derive iterative reconstruction algorithms for reconstruction of f in V from its samples and average samples. We also obtain the error estimates in iterative reconstruction algorithm from noisy samples and iterative noise.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Aldroubi, A., Gröchenig, K.: Beurling–Landau-type theorems for nonuniform sampling in shift-invariant spline spaces. J. Fourier Anal. Appl. 6(1), 93–103 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Aldroubi, A., Gröchenig, K.: Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43(4), 585–620 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Aldroubi, A., Sun, Q., Tang, W.S.: \(p\)-frames and shift invariant subspaces of \(L^p\). J. Fourier Anal. Appl. 7(1), 1–22 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Aldroubi, A., Sun, Q., Tang, W.S.: Convolution, average sampling, and a Calderon resolution of the identity for shift-invariant spaces. J. Fourier Anal. Appl. 11(2), 215–244 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Benedek, A., Panzone, R.: The space \(L^p\) with mixed norm. Duke Math. J. 28(3), 301–324 (1961)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Bhandari, A., Zayed, A.I.: Shift-invariant and sampling spaces associated with the fractional Fourier transform domain. IEEE Trans. Signal Process. 60(4), 1627–1637 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Bhandari, A., Zayed, A.I.: Shift-invariant and sampling spaces associated with the special affine Fourier transform. Appl. Comput. Harmon. Anal. (2017). https://doi.org/10.1016/j.acha.2017.07.002

    MATH  Article  Google Scholar 

  8. 8.

    Butzer, P.L., Stens, R.L.: Sampling theory for not necessarily band-limited functions: a historical overview. SIAM Rev. 34(1), 40–53 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Cheng, C., Jiang, Y., Sun, Q.: Spatially distributed sampling and reconstruction. Appl. Comput. Harmon. Anal. 47, 109–148 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Fernandez, D.L.: Vector-valued singular integral operators on \(L^p\)-spaces with mixed norms and applications. Pac. J. Math. 129(2), 257–275 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Francia, J.L., Ruiz, F.J., Torrea, J.L.: Calderón–Zygmund theory for operator-valued kernels. Adv. Math. 62(1), 7–48 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Gröchenig, K., Romero, J.L., Stöckler, J.: Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions. Invent. math. 211(3), 1119–1148 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Gröchenig, K., Stöckler, J.: Gabor frames and totally positive functions. Duke Math. J. 162(6), 1003–1031 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Jiang, Y., Sun, W.: Adaptive sampling of time-space signals in a reproducing kernel subspace of mixed lebesgue space. arXiv preprint arXiv:1904.00727 (2019)

  15. 15.

    Kulkarni, S.H., Radha, R., Sivananthan, S.: Non-uniform sampling problem. J. Appl. Funct. Anal. 4(1), 58–74 (2009)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Kumar, A., Sampath, S.: Average sampling and reconstruction in shift-invariant spaces and variable bandwidth spaces. Appl. Anal. (2018). https://doi.org/10.1080/00036811.2018.1508652

    MATH  Article  Google Scholar 

  17. 17.

    Li, R., Liu, B., Liu, R., Zhang, Q.: Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue spaces \(L^{p, q}({\mathbb{R}}^{d+1})\). J. Math. Anal. Appl. 453(2), 928–941 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Moumni, T., Zayed, A.I.: A generalization of the prolate spheroidal wave functions with applications to sampling. Integral Transforms Spec. Funct. 25(6), 433–447 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Nashed, M.Z., Sun, Q.: Sampling and reconstruction of signals in a reproducing kernel subspace of \( L^p ({\mathbb{R}}^d)\). J. Funct. Anal. 258(7), 2422–2452 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Nashed, M.Z., Sun, Q., Xian, J.: Convolution sampling and reconstruction of signals in a reproducing kernel subspace. Proc. Am. Math. Soc. 141(6), 1995–2007 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Razafinjatovo, H.N.: Iterative reconstructions in irregular sampling with derivatives. J. Fourier Anal. Appl. 1(3), 281–295 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Samarah, S., Obeidat, S., Salman, R.: A schur test for weighted mixed-norm spaces. Anal. Math. 31(4), 277–289 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Selvan, A.A., Radha, R.: Sampling and reconstruction in shift-invariant spaces of \(B\)-spline functions. Acta Appl. Math. 145(1), 175–192 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Selvan, A.A., Radha, R.: An optimal result for sampling density in shift-invariant spaces generated by Meyer scaling function. J. Math. Anal. Appl. 451(1), 197–208 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Sun, Q.: Nonuniform average sampling and reconstruction of signals with finite rate of innovation. SIAM J. Math. Anal. 38(5), 1389–1422 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Sun, W., Zhou, X.: Reconstruction of band-limited functions from local averages. Constr. Approx. 18(2), 205–222 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Sun, W., Zhou, X.: Reconstruction of functions in spline subspaces from local averages. Proc. Am. Math. Soc. 131(8), 2561–2571 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Tao, R., Li, B.Z., Wang, Y., Aggrey, G.K.: On sampling of band-limited signals associated with the linear canonical transform. IEEE Trans. Signal Process. 56(11), 5454–5464 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Vetterli, M., Marziliano, P., Blu, T.: Sampling signals with finite rate of innovation. IEEE Trans. Signal Process. 50(6), 1417–1428 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Walter, G.: A sampling theorem for wavelet subspaces. IEEE Trans. Inf. Theory 38(2), 881–884 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Walter, G., Shen, X.: Sampling with prolate spheroidal wave functions. Sampl. Theory Signal Image Process. 2(1), 25–52 (2003)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Ward, E.L.: New estimates in harmonic analysis for mixed Lebesgue spaces. Ph.D. thesis, University of Kansas (2010)

  33. 33.

    Wei, D., Ran, Q., Li, Y.: Generalized sampling expansion for band-limited signals associated with the fractional Fourier transform. IEEE Signal Process. Lett. 17(6), 595–598 (2010)

    Article  Google Scholar 

  34. 34.

    Xian, J.: Weighted sampling and reconstruction in weighted reproducing kernel spaces. J. Math. Anal. Appl. 367(1), 34–42 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Xian, J., Luo, S.P., Lin, W.: Weighted sampling and signal reconstruction in spline subspaces. Signal Process. 86(2), 331–340 (2006)

    MATH  Article  Google Scholar 

  36. 36.

    Zayed, A.I.: Sampling of signals band-limited to a disc in the linear canonical transform domain. IEEE Signal Process. Lett. 25(12), 1765–1769 (2018)

    Article  Google Scholar 

  37. 37.

    Zhang, Q.: Nonuniform average sampling in multiply generated shift-invariant subspaces of mixed lebesgue spaces. arXiv preprint arXiv:1806.05055 (2018)

Download references

Acknowledgements

The authors are grateful to the anonymous reviewer for meticulously reading the manuscript, and giving us valuable comments and suggestions which helped to improve the quality of the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sivananthan Sampath.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kumar, A., Patel, D. & Sampath, S. Sampling and reconstruction in reproducing kernel subspaces of mixed Lebesgue spaces. J. Pseudo-Differ. Oper. Appl. 11, 843–868 (2020). https://doi.org/10.1007/s11868-019-00315-0

Download citation

Keywords

  • Nonuniform sampling
  • Reproducing kernel subspaces
  • Shift-invariant spaces
  • Average sampling
  • Reconstruction algorithms
  • Integral operator
  • Mixed Lebesgue spaces

Mathematics Subject Classification

  • 42C15
  • 94A20
  • 47B34
  • 32A70