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Multidimensional Fourier Methods

  • Gerlind Plonka
  • Daniel Potts
  • Gabriele Steidl
  • Manfred Tasche
Chapter
First Online:
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

In this chapter, we consider d-dimensional Fourier methods for fixed \(d\in \mathbb N\). We start with Fourier series of d-variate, 2π-periodic functions \(f:\,\mathbb T^d \to \mathbb C\) in Sect. 4.1, where we follow the lines of Chap.  1. In particular, we present basic properties of the Fourier coefficients and learn about their decay for smooth functions.

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References

  1. 10.
    K. Atkinson, W. Han, Theoretical Numerical Analysis. A Functional Analysis Framework (Springer, Dordrecht, 2009)Google Scholar
  2. 24.
    E. Bedrosian, A product theorem for Hilbert transforms. Proc. IEEE 51(5), 868–869 (1963)CrossRefGoogle Scholar
  3. 42.
    A. Bovik, Handbook of Image and Video Processing, 2nd edn. (Academic, Burlington, 2010)zbMATHGoogle Scholar
  4. 49.
    J.L. Brown, Analytic signals and product theorems for Hilbert transforms. IEEE Trans. Circuits Syst. 21, 790–792 (1974)MathSciNetCrossRefGoogle Scholar
  5. 51.
    T. Bülow, G. Sommer, Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case. IEEE Trans. Signal Process. 49(11), 2844–2852 (2001)MathSciNetCrossRefGoogle Scholar
  6. 109.
    M. Felsberg, G. Sommer, The monogenic signal. IEEE Trans. Signal Process. 49(12), 3136–3144 (2001)MathSciNetCrossRefGoogle Scholar
  7. 115.
    W. Förstner, E. Gülch, A fast operator for detection and precise location of distinct points, corners and centres of circular features, in Proceedings of ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data (1987), pp. 281–305Google Scholar
  8. 129.
    J.E. Gilbert, M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis (Cambridge University Press, Cambridge, 1991)CrossRefGoogle Scholar
  9. 146.
    L. Grafakos, Classical Fourier Analysis, 2nd edn. (Springer, New York, 2008)zbMATHGoogle Scholar
  10. 151.
    K. Gröchenig, An uncertainty principle related to the Poisson summation formula. Stud. Math. 121(1), 87–104 (1996)MathSciNetCrossRefGoogle Scholar
  11. 154.
    S.L. Hahn, Hilbert Transforms in Signal Processing (Artech House, Boston, 1996)zbMATHGoogle Scholar
  12. 158.
    S. Häuser, B. Heise, G. Steidl, Linearized Riesz transform and quasi-monogenic shearlets. Int. J. Wavelets Multiresolut. Inf. Process. 12(3), 1–25 (2014)MathSciNetCrossRefGoogle Scholar
  13. 200.
    F.W. King, Hilbert Transforms, Volume I (Cambridge University Press, Cambridge, 2008)Google Scholar
  14. 201.
    F.W. King, Hilbert Transforms, Volume II (Cambridge University Press, Cambridge, 2009)zbMATHGoogle Scholar
  15. 209.
    U. Köthe, M. Felsberg, Riesz-transforms versus derivatives: on the relationship between the boundary tensor and the energy tensor, in Scale Space and PDE Methods in Computer Vision: 5th International Conference, Scale-Space 2005, ed. by R. Kimmel, N.A. Sochen, J. Weickert (Springer, Berlin, 2005), pp. 179–191CrossRefGoogle Scholar
  16. 220.
    K.G. Larkin, D.J. Bone, M.A. Oldfield, Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform. J. Opt. Soc. Am. A 18(8), 1862–1870 (2001)Google Scholar
  17. 221.
    R. Lasser, Introduction to Fourier Series (Marcel Dekker, New York, 1996)zbMATHGoogle Scholar
  18. 234.
    S. Mallat, A Wavelet Tour of Signal Processing. The Sparse Way, 3rd edn. (Elsevier/Academic, Amsterdam, 2009)Google Scholar
  19. 254.
    H.Q. Nguyen, M. Unser, J.P. Ward, Generalized Poisson summation formulas for continuous functions of polynomial growth. J. Fourier Anal. Appl. 23(2), 442–461 (2017)MathSciNetCrossRefGoogle Scholar
  20. 309.
    M. Riesz, Sur les fonctions conjuguées. Math. Z. 27(1), 218–244 (1928)MathSciNetCrossRefGoogle Scholar
  21. 325.
    L. Schwartz, Théorie des Distributions, (French), nouvelle edn. (Hermann, Paris, 1966)Google Scholar
  22. 329.
    V.L. Shapiro, Fourier Series in Several Variables with Applications to Partial Differential Equations (Chapman & Hall/CRC, Boca Raton, 2011)CrossRefGoogle Scholar
  23. 340.
    E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970)zbMATHGoogle Scholar
  24. 341.
    E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, Princeton, 1971)zbMATHGoogle Scholar
  25. 342.
    M. Storath, Directional multiscale amplitude and phase decomposition by the monogenic curvelet transform. SIAM J. Imag. Sci. 4(1), 57–78 (2011)MathSciNetCrossRefGoogle Scholar
  26. 357.
    H. Triebel, Höhere Analysis (Deutscher Verlag der Wissenschaften, Berlin, 1972)zbMATHGoogle Scholar
  27. 361.
    M. Unser, D. Van De Ville, Wavelet steerability and the higher-order Riesz transform. IEEE Trans. Image Process. 19(3), 636–652 (2010)MathSciNetCrossRefGoogle Scholar
  28. 373.
    H. Wendland, Scattered Data Approximation (Cambridge University Press, Cambridge, 2005)zbMATHGoogle Scholar
  29. 383.
    Y. Xu, D. Yan, The Bedrosian identity for the Hilbert transform of product functions. Proc. Am. Math. Soc. 134(9), 2719–2728 (2006)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Gerlind Plonka
    • 1
  • Daniel Potts
    • 2
  • Gabriele Steidl
    • 3
  • Manfred Tasche
    • 4
  1. 1.University of GöttingenGöttingenGermany
  2. 2.Chemnitz University of TechnologyChemnitzGermany
  3. 3.TU KaiserslauternKaiserslauternGermany
  4. 4.University of RostockRostockGermany

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