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Numerical Fourier Analysis pp 159–230Cite as

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Multidimensional Fourier Methods

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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. K. Atkinson, W. Han, Theoretical Numerical Analysis. A Functional Analysis Framework (Springer, Dordrecht, 2009)

    Google Scholar 

  2. E. Bedrosian, A product theorem for Hilbert transforms. Proc. IEEE 51(5), 868–869 (1963)

    Article  Google Scholar 

  3. A. Bovik, Handbook of Image and Video Processing, 2nd edn. (Academic, Burlington, 2010)

    MATH  Google Scholar 

  4. J.L. Brown, Analytic signals and product theorems for Hilbert transforms. IEEE Trans. Circuits Syst. 21, 790–792 (1974)

    Article  MathSciNet  Google Scholar 

  5. 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)

    Article  MathSciNet  Google Scholar 

  6. M. Felsberg, G. Sommer, The monogenic signal. IEEE Trans. Signal Process. 49(12), 3136–3144 (2001)

    Article  MathSciNet  Google Scholar 

  7. 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. 281–305

    Google Scholar 

  8. J.E. Gilbert, M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis (Cambridge University Press, Cambridge, 1991)

    Book  Google Scholar 

  9. L. Grafakos, Classical Fourier Analysis, 2nd edn. (Springer, New York, 2008)

    MATH  Google Scholar 

  10. K. Gröchenig, An uncertainty principle related to the Poisson summation formula. Stud. Math. 121(1), 87–104 (1996)

    Article  MathSciNet  Google Scholar 

  11. S.L. Hahn, Hilbert Transforms in Signal Processing (Artech House, Boston, 1996)

    MATH  Google Scholar 

  12. S. Häuser, B. Heise, G. Steidl, Linearized Riesz transform and quasi-monogenic shearlets. Int. J. Wavelets Multiresolut. Inf. Process. 12(3), 1–25 (2014)

    Article  MathSciNet  Google Scholar 

  13. F.W. King, Hilbert Transforms, Volume I (Cambridge University Press, Cambridge, 2008)

    Google Scholar 

  14. F.W. King, Hilbert Transforms, Volume II (Cambridge University Press, Cambridge, 2009)

    MATH  Google Scholar 

  15. 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. 179–191

    Chapter  Google Scholar 

  16. 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. R. Lasser, Introduction to Fourier Series (Marcel Dekker, New York, 1996)

    MATH  Google Scholar 

  18. S. Mallat, A Wavelet Tour of Signal Processing. The Sparse Way, 3rd edn. (Elsevier/Academic, Amsterdam, 2009)

    Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. M. Riesz, Sur les fonctions conjuguées. Math. Z. 27(1), 218–244 (1928)

    Article  MathSciNet  Google Scholar 

  21. L. Schwartz, Théorie des Distributions, (French), nouvelle edn. (Hermann, Paris, 1966)

    Google Scholar 

  22. V.L. Shapiro, Fourier Series in Several Variables with Applications to Partial Differential Equations (Chapman & Hall/CRC, Boca Raton, 2011)

    Book  Google Scholar 

  23. E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970)

    MATH  Google Scholar 

  24. E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, Princeton, 1971)

    MATH  Google Scholar 

  25. M. Storath, Directional multiscale amplitude and phase decomposition by the monogenic curvelet transform. SIAM J. Imag. Sci. 4(1), 57–78 (2011)

    Article  MathSciNet  Google Scholar 

  26. H. Triebel, Höhere Analysis (Deutscher Verlag der Wissenschaften, Berlin, 1972)

    MATH  Google Scholar 

  27. M. Unser, D. Van De Ville, Wavelet steerability and the higher-order Riesz transform. IEEE Trans. Image Process. 19(3), 636–652 (2010)

    Article  MathSciNet  Google Scholar 

  28. H. Wendland, Scattered Data Approximation (Cambridge University Press, Cambridge, 2005)

    MATH  Google Scholar 

  29. Y. Xu, D. Yan, The Bedrosian identity for the Hilbert transform of product functions. Proc. Am. Math. Soc. 134(9), 2719–2728 (2006)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Göttingen, Göttingen, Germany

    Gerlind Plonka

  2. Chemnitz University of Technology, Chemnitz, Germany

    Daniel Potts

  3. TU Kaiserslautern, Kaiserslautern, Germany

    Gabriele Steidl

  4. University of Rostock, Rostock, Germany

    Manfred Tasche

Authors
  1. Gerlind Plonka
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  2. Daniel Potts
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  3. Gabriele Steidl
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  4. Manfred Tasche
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Plonka, G., Potts, D., Steidl, G., Tasche, M. (2018). Multidimensional Fourier Methods. In: Numerical Fourier Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-04306-3_4

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