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Acta Applicandae Mathematicae

, Volume 145, Issue 1, pp 175–192 | Cite as

Sampling and Reconstruction in Shift Invariant Spaces of \(B\)-Spline Functions

  • A. Antony Selvan
  • R. Radha
Article

Abstract

A Kadec-type theorem is proved for functions belonging to the shift invariant space \(V(Q_{r})\), where \(Q_{r}\) denotes the \(B\)-spline function of even order \(r\). It is also shown that if a non-zero function \(f\in V(Q_{m})\) has infinitely many simple or double zeros on the real axis which are separated, then there exists at least one pair of consecutive zeros whose distance apart is greater than or equal to a certain number which depends on Krein–Favard constants. Further, iterative reconstruction algorithms are provided for functions in \(V(Q_{m})\). Finally, a sampling density theorem for irregular Gabor-type frames \(\{M_{y_{j}}T_{x_{i,j}}g^{(l)}:l=0, 1, \dots , {k-1}, i, j \in \mathbb{Z} \}\) is proved for \(g\in V(Q_{m})\cap L^{1}( \mathbb{R})\) under certain sufficient conditions on \(g\).

Keywords

B-Splines Bernstein’s inequality Frames Hermite interpolation Krein–Favard constants Nonuniform sampling Riesz basis Wirtinger–Sobolev inequality 

Mathematics Subject Classification

65D07 94A20 

Notes

Acknowledgements

We wish to thank the referees for meticulously reading the manuscript and giving us several valuable suggestions to improve our earlier version of the manuscript.

References

  1. 1.
    Aldroubi, A., Gröchenig, K.: Buerling–Landau-type theorems for non-uniform sampling in shift invariant spline spaces. J. Fourier Anal. Appl. 6(1), 93–103 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aldroubi, A., Gröchenig, K.: Non-uniform sampling and reconstruction in shift invariant subspaces. SIAM Rev. 43(4), 585–620 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Antony Selvan, A., Radha, R.: Sampling and reconstruction in shift invariant spaces on \(\mathbb{R}^{d}\). Ann. Mat. Pura Appl. (4) 194(6), 1683–1706 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Antony Selvan, A., Radha, R.: Separation of zeros and a Hermite interpolation based frame algorithm for band limited functions. arXiv:1511.04007v1 [math.CA]
  5. 5.
    Babenko, V.F., Zontov, V.A.: Bernstein-type inequalities for splines defined on the real axis. Ukr. Math. J. 63(5), 699–708 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Böttcher, A., Widom, H.: From Toeplitz eigenvalues through Green’s kernel to higher-order Wirtinger–Sobolev inequalities. Oper. Theory, Adv. Appl. 171, 73–87 (2007) CrossRefzbMATHGoogle Scholar
  7. 7.
    Chui, C.K.: An Introduction to Wavelets. Academic Press, Boston (1992) zbMATHGoogle Scholar
  8. 8.
    Duffin, R.J., Eachus, J.J.: Some notes on an expansion theorem of Paley and Wiener. Bull. Am. Math. Soc. 48, 850–855 (1942) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dym, H., Mckean, H.P.: Fourier Series and Integrals. Academic Press, New York (1972) zbMATHGoogle Scholar
  10. 10.
    Feichtinger, H.G.: Banach spaces of distributions of Wiener’s type and interpolation. In: Functional Analysis and Approximation, Oberwolfach, 1980. Internat. Ser. Numer. Math., vol. 60. Birkhäuser, Basel (1981) Google Scholar
  11. 11.
    Feichtinger, H.G., Gröchenig, K.: Theory and practice of irregular sampling. In: Benedetto, J.J., Frazier, M.W. (eds.) Wavelets: Mathematics and Applications. Studies in Advanced Mathematics. CRC Press, Boca Raton (1993) Google Scholar
  12. 12.
    Gocheva-Ilieva, S.G., Feschiev, I.H.: New recursive representations for the Favard constants with applications to multiple singular integrals and summation of series. Abstr. Appl. Anal. 2013, 523618 (2013), 12 pp. MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gröchenig, K.: Reconstruction algorithms in irregular sampling. Math. Comput. 59(199), 181–194 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gröchenig, K.: Irregular sampling of wavelet and short-time Fourier transforms. Constr. Approx. 9, 283–297 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kadec, M.I.: The exact value of Paley–Wiener constants. Sov. Math. Dokl. 5, 559–561 (1964) MathSciNetGoogle Scholar
  16. 16.
    Landau, H.J.: Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math. 117, 37–52 (1967) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Liu, Y.: Irregular sampling for spline wavelet subspaces. IEEE Trans. Inf. Theory 42, 623–627 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liu, Y.M., Walter, G.G.: Irregular sampling in wavelet subspaces. J. Fourier Anal. Appl. 2(2), 181–189 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mischenko, E.V.: Determination of Riesz bounds for the spline basis with the help of trigonometric polynomials. Sib. Math. J. 51(4), 660–666 (2010) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Paley, R.E.A.C., Wiener, N.: Fourier Transforms in the Complex Domain. Am. Math. Soc., Providence (1934) zbMATHGoogle Scholar
  21. 21.
    Razafinjatovo, H.N.: Iterative reconstructions in irregular sampling with derivatives. J. Fourier Anal. Appl. 1(3), 281–295 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schoenberg, I.J.: Cardinal Spline Interpolation. CBMS-NSF Series in Applied Math., vol. 12. SIAM, Philadelphia (1973) CrossRefzbMATHGoogle Scholar
  23. 23.
    Spitzbart, A.: A generalization of Hermite’s interpolation formula. Am. Math. Mon. 67(1), 42–46 (1960) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sun, W., Zhou, X.: Reconstruction of functions in spline subspace from local averages. Proc. Am. Math. Soc. 131(8), 2561–2571 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Walker, W.J.: The separation of zeros for entire functions of exponential type. J. Math. Anal. Appl. 122, 257–259 (1987) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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