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On multi-dimensional sampling and interpolation

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Abstract

The paper discusses sharp sufficient conditions for interpolation and sampling for functions of n variables with convex spectrum. When n = 1, the classical theorems of Ingham and Beurling state that the critical values in the estimates from above (from below) for the distances between interpolation (sampling) nodes are the same. This is no longer true for n > 1. While the critical value for sampling sets remains constant, the one for interpolation grows linearly with the dimension.

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References

  1. Baiocchi C., Komornik V., Loreti P.: Théorèmes du type Ingham et application á la thèorie du contrôle. C. R. Acad. Sci. Paris Sér. I Math. 326(4), 453–458 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benedetto, J.J., Wu, H.-Ch.: Non-uniform sampling and spiral MRI reconstruction. In: SPIE-Wavelet Applications in Signal and Image Processing VIII, vol. 4119, pp. 130–141 (2000)

  3. Beurling, A.: Interpolation for an interval in \({\mathbb{R}^1}\) . In: The collected Works of Arne Beurling. Harmonic Analysis, vol. 2. Birkhauser, Boston (1989)

  4. Beurling, A.: Balayage of Fourier–Stieltjes transforms. In: The collected Works of Arne Beurling. Harmonic Analysis, vol. 2. Birkhauser, Boston (1989)

  5. Beurling, A.: Local Harmonic Analysis with some Applications to Differential Operators. In: The collected Works of Arne Beurling. Harmonic Analysis, vol. 2. Birkhauser, Boston (1989)

  6. Bezuglaya L., Katsnelson V.: The sampling theorem for functions with limited multi-band spectrum. Z. Anal. Anwendungen 12(3), 511–534 (1993)

    MathSciNet  MATH  Google Scholar 

  7. Cassels J.W.S.: An Introduction to the Geometry of Numbers. Springer, Berlin (1971)

    MATH  Google Scholar 

  8. Clunie J., Rahman Q.I., Walker W.J.: On entire functions of exponential type bounded on the real axis. J. Lond. Math. Soc. (2) 61(1), 163–176 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duffin R.J., Schaeffer A.C.: Some properties of functions of exponential type. Bull. Am. Math. Soc. 44, 236–240 (1938)

    Article  MathSciNet  Google Scholar 

  10. Higgins J.R.: Sampling Theory in Fourier and Signal Analysis. Foundations. Clarendon Press, Oxford (1996)

    MATH  Google Scholar 

  11. Hörmander L.: Some inequalities for functions of exponential type. Math. Scand. 3, 21–27 (1955)

    MathSciNet  MATH  Google Scholar 

  12. Ingham A.E.: Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41, 367–379 (1936)

    Article  MathSciNet  Google Scholar 

  13. Kahane J.-P.: Sur les fonctions moyenne-périodiques bornées. Ann. Inst. Fourier 7, 293–314 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kahane, J.-P.: Fonctions pseudo-périodiques dans \({\mathbb{R}^{p}}\) (French). In: Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pp. 274–281. Jerusalem Academic Press/Pergamon, Jerusalem/Oxford (1961)

  15. Kahane J.-P.: Pseudopériodicité et séries de Fourier lacunaires. Ann. Sci. Ecole Norm. Sup. 79, 93–150 (1962)

    MathSciNet  MATH  Google Scholar 

  16. Komornik V., Loreti P.: Fourier series in control theory. Springer Monographs in Mathematics. Springer, Berlin (2005)

    Google Scholar 

  17. Landau H.J.: A sparse regular sequence of exponentials closed on large sets. Bull. Am. Math. Soc. 70, 566–569 (1964)

    Article  MATH  Google Scholar 

  18. Landau H.J.: Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math. 117, 37–52 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  19. Levin B.Ya.: Lectures on Entire Functions. AMS, Providence (1996)

    MATH  Google Scholar 

  20. Lyubarskii Yu., Rashkovskii A.: Complete interpolating sequences for Fourier transforms supported by convex symmetric polygons. Ark. Mat. 38(1), 139–170 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Matei B., Meyer Y.: Quasicrystals are sets of stable sampling. C. R. Math. Acad. Sci. Paris 346(23–24), 1235–1238 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Matei B., Meyer Y.: A variant of compressed sensing. Rev. Mat. Iberoam 25(2), 669–692 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Matei B., Meyer Y.: Simple quasicrystals are sets of stable sampling. Complex Var. Elliptic Equ. 55(8–10), 947–964 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Olevskii A., Ulanovskii A.: Universal sampling of band-limited signals. C. R. Math. Acad. Sci. Paris 342(12), 927–931 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Olevskii A., Ulanovskii A.: Universal sampling and interpolation of band-limited signals. Geom. Funct. Anal. 18(3), 1029–1052 (2008)

    Article  MathSciNet  Google Scholar 

  26. Olevskii A., Ulanovskii A.: On Ingham-type interpolation in \({\mathbb{R}^n}\) . C. R. Math. Acad. Sci. Paris 348(13–14), 807–810 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rogers C.A.: Packing and Covering. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  28. Seip, K.: Interpolation and sampling in spaces of analytic functions. In: University Lecture Series, vol. 33. American Mathematical Society, Providence (2004)

  29. Stein E., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1987)

    Google Scholar 

  30. Ulanovskii A.: Sparse systems of functions closed on large sets in \({\mathbf{R}^N}\) . J. Lond. Math. Soc. (2) 63(2), 428–440 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Young R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, London (2001)

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel

    Alexander Olevskii

  2. Stavanger University, 4036, Stavanger, Norway

    Alexander Ulanovskii

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  1. Alexander Olevskii
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  2. Alexander Ulanovskii
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Correspondence to Alexander Olevskii.

Additional information

A. Olevskii is supported in part by the Israel Science Foundation and A. Ulanovskii is supported by an ESF-HCAA grant.

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Olevskii, A., Ulanovskii, A. On multi-dimensional sampling and interpolation. Anal.Math.Phys. 2, 149–170 (2012). https://doi.org/10.1007/s13324-012-0027-4

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  • DOI: https://doi.org/10.1007/s13324-012-0027-4

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