Skip to main content
SpringerLink
Log in

Fourier Operators in Applied Harmonic Analysis

  • Chapter
Sampling Theory, a Renaissance

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

  • 2144 Accesses

Abstract

We present a panorama describing the pervasiveness of the short-time Fourier transform (STFT) in a host of topics including the following: waveform design and optimal ambiguity function behavior for radar and communications applications; vector-valued ambiguity function theory for multi-sensor environments; finite Gabor frames for deterministic compressive sensing and as a background for the HRT conjecture; generalizations of Fourier frames and non-uniform sampling; and pseudo-differential operator frame inequalities.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
eBook
USD 84.99
Price excludes VAT (USA)
Softcover Book
USD 109.99
Price excludes VAT (USA)
Hardcover Book
USD 109.99
Price excludes VAT (USA)

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. S.T. Ali, J.-P. Antoine, J.-P. Gazeau, Continuous frames in Hilbert space. Ann. Phys. 222, 1–37 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. S.T. Ali, J.-P. Antoine, J.-P. Gazeau, Coherent States, Wavelets and Their Generalizations. Graduate Text in Contemporary Physics (Springer, New York, 2000)

    Google Scholar 

  3. W.O. Alltop, Complex sequences with low periodic correlations. IEEE Trans. Inf. Theory 26(3), 350–354 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. T. Andrews, Representations of finite groups for frame representation theory (submitted)

    Google Scholar 

  5. T. Andrews, J.J. Benedetto, J.J. Donatelli, Frame multiplication theory and vector-valued ambiguity functions (submitted)

    Google Scholar 

  6. E. Au-Yeung, J.J. Benedetto, Generalized Fourier frames in terms of balayage (submitted)

    Google Scholar 

  7. L. Auslander, P.E. Barbano, Communication codes and Bernoulli transformations. Appl. Comput. Harmon. Anal. 5(2), 109–128 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Auslander, I. Gertner, Wide-band ambiguity function and ax+ b group. Inst. Math. Its Appl. 22, 1 (1990)

    Article  MathSciNet  Google Scholar 

  9. R. Balan, A noncommutative Wiener lemma and a faithful tracheal state on Banach algebra of time-frequency operators. Trans. Am. Math. Soc. 360, 3921–3941 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. P.A. Bello, Measurement of random time-variant linear channels. IEEE Trans. Inf. Theory 15(4), 469–475 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  11. J.J. Benedetto, Harmonic Analysis on Totally Disconnected Sets. Springer Lecture Notes, vol. 202 (Springer, New York, 1971)

    Google Scholar 

  12. J.J. Benedetto, Spectral Synthesis (Academic, New York, 1975)

    Book  Google Scholar 

  13. J.J. Benedetto, Harmonic Analysis and Applications (CRC Press, Boca Raton 1997)

    Google Scholar 

  14. J.J. Benedetto, Sampling, Sparsity, and CAZAC Sequences. Lecture Notes in Applied and Numerical Harmonic Analysis (Springer/Birkhäuser, New York, 2015)

    Google Scholar 

  15. J.J. Benedetto, A. Bourouihiya, Lineaer independence of finite Gabor systems determined by behavior at infinity. J. Geom. Anal. (2014)

    Google Scholar 

  16. J.J. Benedetto, W. Czaja, Integration and Modern Analysis. Birkhäuser Advanced Texts (Springer/Birkhäuser, New York, 2009)

    Book  MATH  Google Scholar 

  17. J.J. Benedetto, S. Datta, Construction of infinite unimodular sequences with zero autocorrelation. Adv. Comput. Math. 32, 191–207 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. J.J. Benedetto, J.J. Donatelli, Ambiguity function and frame theoretic properties of periodic zero autocorrelation waveforms. IEEE J. Spec. Top Signal Process. 1, 6–20 (2007)

    Article  Google Scholar 

  19. J.J. Benedetto, J.J. Donatelli, Frames and a vector-valued ambiguity function, in Asilomar Conference on Signals, Systems, and Computers (Oct 2008)

    Google Scholar 

  20. J.J. Benedetto, A. Nava-Tudela, Sampling in image representation and compression, in Sampling Theory in Honor of Paul L. Bützer’s 85th Birthday (Springer/Birkhäuser, New York, 2014)

    Google Scholar 

  21. J.J. Benedetto, I. Konstantinidis, M. Rangaswamy, Phase-coded waveforms and their design. IEEE Signal Process. Mag. 26(1), 22–31 (2009)

    Article  Google Scholar 

  22. J.J. Benedetto, R.L. Benedetto, J.T. Woodworth, Optimal ambiguity functions and Weil’s exponential sum bound. J. Fourier Anal. Appl. 18(3), 471–487 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Beurling, Local harmonic analysis with some applications to differential operators, in Some Recent Advances in the Basic Sciences, vol. 1 (Proceedings Annual Science Conference, Belfer Graduate School of Science, Yeshiva University, New York, 1962–1964), pp. 109–125

    Google Scholar 

  24. A. Beurling, The Collected Works of Arne Beurling. Vol. 2. Harmonic Analysis (Springer/Birkhäuser, New York, 1989)

    Google Scholar 

  25. A. Beurling, P. Malliavin, On Fourier transforms of measures with compact support. Acta Math. 107, 291–309 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Beurling, P. Malliavin, On the closure of characters and the zeros of entire functions. Acta Math. 118, 79–93 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  27. G. Björck, Functions of modulus one on Z p whose Fourier transforms have constant modulus, in Proceedings of the A. Haar Memorial Conference, vol. I, II (Budapest, 1985). Colloquim Mathematical Society János Bolyai, vol. 49 (North-Holland, Amsterdam, 1987), pp. 193–197

    Google Scholar 

  28. G. Björck, Functions of modulus one on \(\mathbb{Z}_{n}\) whose Fourier transforms have constant modulus, and cyclic n-roots, in Proceedings of 1989 NATO Advanced Study Institute on Recent Advances in Fourier Analysis and its Applications (1990), pp. 131–140

    Google Scholar 

  29. G. Björck, B. Saffari, New classes of finite unimodular sequences with unimodular Fourier transforms. Circulant Hadamard matrices with complex entries. C. R. Acad. Sci. Paris 320, 319–324 (1995)

    MATH  Google Scholar 

  30. H. Bölcskei, Y.C. Eldar, Geometrically uniform frames. IEEE Trans. Inf. Theory 49(4), 993–1006 (2003)

    Article  MATH  Google Scholar 

  31. A.M Bruckstein, D.L Donoho, M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51(1), 34–81 (2009)

    Google Scholar 

  32. P.L. Butzer, F. Fehër, E. B. Christoffel - The Influence of his Work on Mathematics and the Physical Sciences (Springer/Birkhäuser, New York, 1981)

    Google Scholar 

  33. E.J. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)

    Article  MATH  Google Scholar 

  34. O. Christensen, An Introduction to Frames and Riesz Bases (Springer/Birkhäuser, New York, 2003)

    Book  MATH  Google Scholar 

  35. D.C. Chu, Polyphase codes with good periodic correlation properties. IEEE Trans. Inf. Theory 18, 531–532 (1972)

    Article  MATH  Google Scholar 

  36. L. Cohen, Time-Frequency Analysis: Theory and Applications (Prentice-Hall Inc, Upper Saddle River, 1995)

    Google Scholar 

  37. C. Demeter, Linear independence of time frequency translates for special configurations. Math. Res. Lett. 17, 761–799 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. C. Demeter, A. Zaharescu, Proof of the HRT conjecture for (2, 2) configurations. J. Math. Anal. Appl. 388(1), 151–159 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  40. H.G. Feichtinger, On a new Segal algebra. Monatsh. Math. 92, 269–289 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  41. H.G. Feichtinger, Modulation spaces on locally compact abelian groups, in Proceedings of International Conference on Wavelets and Applications (1983), pp. 1–56

    Google Scholar 

  42. H.G. Feichtinger, Wiener amalgams over Euclidean spaces and some of their applications, in Proceedings Conference of Function Spaces (Edwardsville, IL, 1990). Lecture Notes in Pure and Applied Mathematics, vol. 136 (1990), pp. 123–137

    MathSciNet  Google Scholar 

  43. H.G. Feichtinger, K.H. Gröchenig, A unified approach to atomic decompositions via integrable group representations, in Function Spaces and Applications (Lund 1986). Lecture Notes in Mathematics, vol. 1302 (Springer, Berlin, 1988), pp. 52–73

    Google Scholar 

  44. H.G. Feichtinger, K.H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. J. Funct. Anal. 86, 307–340 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  45. M. Fornasier, H. Rauhut, Continuous frames, function spaces, and the discretization problem. J. Fourier Anal. Appl. 11, 245–287 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  46. G.D. Forney, Geometrically uniform codes. IEEE Trans. Inf. Theory 37(5), 1241–1260 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  47. R.L. Frank, S.A. Zadoff, Phase shift pulse codes with good periodic correlation properties. IRE Trans. Inf. Theory 8, 381–382 (1962)

    Article  Google Scholar 

  48. J.-P. Gabardo, D. Han, Frames associated with measurable spaces. frames. Adv. Comput. Math. 18, 127–147 (2003)

    Google Scholar 

  49. S.W. Golomb, G. Gong, Signal Design for Good Correlation (Cambridge University Press, Cambridge, 2005)

    Book  MATH  Google Scholar 

  50. K.H. Gröchenig, Describing functions: atomic decompositions versus frames. Monatsh. Math. 112, 1–42 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  51. K.H. Gröchenig, Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis (Springer/Birkhäuser, New York, 2001)

    Book  MATH  Google Scholar 

  52. K.H. Gröchenig, A pedestrian’s approach to pseudodifferential operators, in Harmonic Analysis and Applications, ed. by C. Heil (Springer/Birkhäuser, New York, 2006), pp. 139–169

    Chapter  Google Scholar 

  53. J.-C. Guey, M.R. Bell, Diversity waveform sets for delay-doppler imaging. IEEE Trans. Inf. Theory 44(4), 1504–1522 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  54. D. Han, Classification of finite group-frames and super-frames. Can. Math. Bull. 50(1), 85–96 (2007)

    Article  MATH  Google Scholar 

  55. D. Han, D. Larson, Frames, bases and group representations. Mem. Am. Math. Soc. 147(697) (2000)

    Google Scholar 

  56. G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 4th edn. (Oxford University, Oxford, 1965)

    Google Scholar 

  57. C. Heil, Linear independence of finite Gabor systems, in Harmonic Analysis and Applications: A Volume in Honor of John J. Benedetto (Springer/Birkhäuser, New York, 2006)

    Google Scholar 

  58. C. Heil, J. Ramanathan, P. Topiwala, Linear independence of time-frequency translates. Proc. Am. Math. Soc. 124(9), 2787–2795 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  59. T. Helleseth, P. Vijay Kumar, Sequences with low correlation, in Handbook of Coding Theory, vol. I, II, ed. by V.S. Pless, W.C. Huffman (North-Holland, Amsterdam, 1998), pp. 1765–1853

    Google Scholar 

  60. M.A. Herman, T. Strohmer, High-resolution radar via compressed sensing. IEEE Trans. Signal Process. 57(6), 2275–2284 (2009)

    Article  MathSciNet  Google Scholar 

  61. M.J. Hirn, The number of harmonic frames of prime order. Linear Algebra Appl. 432(5), 1105–1125 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  62. L.Hörmander, The Weyl calculus of pseudo differential operators. Commun. Pure Appl. Math. 32, 360–444 (1979)

    Article  Google Scholar 

  63. S. Jaffard, A density criterion for frames of complex exponentials. Mich. Math. J. 38, 339–348 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  64. P. Jaming, Phase retrieval techniques for radar ambiguity problems. J. Fourier Anal. Appl. 5(4), 309–329 (1999) MR 1700086 (2000g:94007)

    Google Scholar 

  65. J.-P. Kahane, Sur certaines classes de séries de Fourier absolument convergentes. J. Math. Pures Appl. (9) 35, 249–259 (1956)

    Google Scholar 

  66. J.-P. Kahane, Séries de Fourier Absolument Convergentes, (Springer, New York, 1970)

    MATH  Google Scholar 

  67. J.-P. Kahane, R. Salem, Ensembles Parfaits et Séries Trigonométriques (Paris, 1963)

    Google Scholar 

  68. T. Kailath, Sampling models for linear time-variant filters. Technical Report 352, Massachusetts Institute of Technology, Research Laboratory of Electronics (1959)

    Google Scholar 

  69. T. Kailath, Measurements on time-variant communication channels. IRE Trans. Inf. Theory 8(5), 229–236 (1962)

    Article  MATH  Google Scholar 

  70. T. Kailath, Time-variant communication channels. IEEE Trans. Inf. Theory 9(4), 233–237 (1963)

    Article  Google Scholar 

  71. Y. Katznelson, An Introduction to Harmonic Analysis, 3rd Original 1968 edn., Cambridge Mathematical Library (Cambridge University Press, Cambridge, 2004)

    Google Scholar 

  72. J.R. Klauder, The design of radar signals having both high range resolution and high velocity resolution. Bell Syst. Tech. J. 39, 809–820 (1960)

    Article  Google Scholar 

  73. J.R. Klauder, A.C. Price, S. Darlington, W.J. Albersheim, The theory and design of chirp radars. Bell Syst. Tech. J. 39, 745–808 (1960)

    Article  Google Scholar 

  74. J.F. Koksma, The theory of asymptotic distribution modulo one. Compos. Math. 16, 1–22 (1964)

    MathSciNet  MATH  Google Scholar 

  75. J. Kovačević, A. Chebira, Life beyond bases: the advent of frames (part I). IEEE Signal Process. Mag. 24(4), 86–104 (2007)

    Article  Google Scholar 

  76. J. Kovačević, A. Chebira, Life beyond bases: the advent of frames (part II). IEEE Signal Process. Mag. 24, 115–125 (2007)

    Google Scholar 

  77. W. Kozek, G.E. Pfander, Identification of operators with bandlimited symbols. SIAM J. Math. Anal. 37(3), 867–888 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  78. G. Kutyniok, Linear independence of time-frequency shifts under a generalized Schrödinger representation. Arch. Math. 78, 135–144 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  80. N. Levanon, E. Mozeson, Radar Signals (Wiley Interscience/IEEE Press, Hoboken, New Jersey, 2004)

    Book  Google Scholar 

  81. L.Ê. Lindahl, F. Poulsen, Thin Sets in Harmonic Analysis: Seminars Held at Institute Mittag-Leffler, 1969/70 (M. Dekker, New York, 1971)

    MATH  Google Scholar 

  82. P.A. Linnell, Von Neumann algebras and linear independence of translates. Proc. Am. Math. Soc. 127, 3269–3277 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  83. Y. Meyer, Algebraic Numbers and Harmonic Analysis (Elsevier, New York, 1972)

    MATH  Google Scholar 

  84. W.H. Mow, A new unified construction of perfect root-of-unity sequences, in Proceedings of IEEE 4th International Symposium on Spread Spectrum Techniques and Applications (Germany) (September 1996), pp. 955–959

    Google Scholar 

  85. J. Ortega-Cerdà, K. Seip, Fourier frames. Ann. Math. 155(3), 789–806 (2002)

    Article  MATH  Google Scholar 

  86. R.E.A.C. Paley, N. Wiener, Fourier Transforms in the Complex Domain. American Mathematical Society Colloquium Publications, vol. XIX (American Mathematical Society, Providence, RI, 1934)

    Google Scholar 

  87. G.E. Pfander, Gabor frames in finite dimensions, in Finite Frames, Theory and Applications, ed. by P.G. Casazza, G. Kutyniok (Springer/Birkhäuser, New York, 2013), pp. 193–239

    Chapter  Google Scholar 

  88. G.E. Pfander, Sampling of operators. J. Fourier Anal. Appl. 19(3), 612–650 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  89. G.E. Pfander, D.F. Walnut, Measurement of time-variant linear channels. IEEE Trans. Inf. Theory 52(11), 4808–4820 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  90. G.E. Pfander, D.F. Walnut, Operator identification and Feichtinger’s algebra. Sampling Theory Signal Image Process. 5(2), 183–200 (2006)

    MathSciNet  MATH  Google Scholar 

  91. B.M. Popovic, Generalized chirp-like polyphase sequences with optimum correlation properties. IEEE Trans. Inf. Theory 38(4), 1406–1409 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  92. R.A. Rankin, The closest packing of spherical caps in n dimensions, in Proceedings of the Glasgow Mathematical Association, vol. 2 (Cambridge University Press, Cambridge, 1955), pp. 139–144

    Google Scholar 

  93. M.A. Richards, J.A. Scheer, W.A. Holm (eds.), Principles of Modern Radar (SciTech Publishing Inc., Raleigh, 2010)

    Google Scholar 

  94. F. Riesz, B. Sz.-Nagy, Functional Analysis (Frederick Ungar Publishing Co., New York, 1955)

    Google Scholar 

  95. W. Rudin, Fourier Analysis on Groups. Interscience Tracts in Pure and Applied Mathematics (Interscience Publishers, New York, 1962)

    Google Scholar 

  96. B. Saffari, Some polynomial extremal problems which emerged in the twentieth century, in Twentieth Century Harmonic Analysis—A Celebration. NATO Science Series II Mathematics, Physics and Chemistry. vol. 33, (Kluwer Academic Publishers, Dordrecht, 2001) pp. 201–233.

    Google Scholar 

  97. R. Salem, On singular monotonic functions of the Cantor type. J. Math. Phys. 21, 69–82 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  98. R. Salem, On singular monotonic functions whose spectrum has a given Hausdorff dimension. Ark. Mat. 1(4), 353–365 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  99. K. Seip, On the connection between exponential bases and certain related sequences in L2(−π, π). J. Funct. Anal. 130, 131–160 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  100. M.I. Skolnik, Introduction to Radar Systems (McGraw-Hill Book Company, New York, 1980)

    Google Scholar 

  101. D. Slepian, Group codes for the gaussian channel. Bell Syst. Tech. J. 47(4), 575–602 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  102. E.M. Stein, Harmonic Analysis (Princeton University Press, Princeton, 1993)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  104. T. Strohmer, R.W. Heath Jr., Grassmannian frames with applications to coding and communications. Appl. Comput. Harmon. Anal. 14, 257–275 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  105. W. Sun, G-frames and g-Riesz bases. J. Math. Anal. Appl. 322(1), 437–452 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  106. D.A. Swick, A review of wideband ambiguity functions. Technical Report, DTIC Document (1969)

    Google Scholar 

  107. A. Terras, Fourier Analysis on Finite Groups and Applications, vol. 43 (Cambridge University Press, Cambridge, 1999)

    MATH  Google Scholar 

  108. R.J. Turyn, Sequences with small correlation, in Error Correcting Codes, (Wiley, New York, 1968), pp. 195–228

    Google Scholar 

  109. D.E. Vakman, Sophisticated Signals and the Uncertainty Principle in Radar (Springer, New York, 1969)

    Google Scholar 

  110. R. Vale, S. Waldron, Tight frames and their symmetries. Constr. Approx. 21(1), 83–112 (2004)

    MathSciNet  Google Scholar 

  111. R. Vale, S. Waldron, Tight frames generated by finite nonabelian groups. Numer. Algorithms 48(1–3), 11–27 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  112. A. Weil, On some exponential sums. Proc. Natl. Acad. Sci. USA 34, 204–207 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  113. A. Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Actualités Sci. et Ind. no. 1041 (Hermann, Paris, 1948)

    Google Scholar 

  114. L. Welch, Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inf. Theory 20(3), 397–399 (1974)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first named author gratefully acknowledges the support of MURI-ARO Grant W911NF- 09-1-0383, NGA Grant 1582-08-1-0009, and DTRA Grant HDTRA1-13-1-0015. The second named author gratefully acknowledges the support of the Norbert Wiener Center at the University of Maryland, College Park.

Author information

Authors and Affiliations

  1. Department of Mathematics, Norbert Wiener Center, University of Maryland, College Park, MD, 20742, USA

    John J. Benedetto & Matthew J. Begué

Authors
  1. John J. Benedetto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthew J. Begué
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John J. Benedetto .

Editor information

Editors and Affiliations

  1. School of Engineering and Science, Jacobs University Bremen, Bremen, Bremen, Germany

    Götz E. Pfander

Appendix

Appendix

The Classical Sampling Theorem goes back to papers by Cauchy (1840s), see [13, Theorem 3.10.10] for proofs of Theorem 17. It has had a significant impact on various topics in mathematics, including number theory and interpolation theory, long before Shannon’s application of it in communications.

Theorem 17 (Classical Sampling Theorem).

Let T,Ω > 0 satisfy the condition that 0 < 2TΩ ≤ 1, and let s be an element of the Paley-Wiener space \(PW_{1/(2T)}\) satisfying the condition that \(\hat{s} = S = 1\) on [−Ω,Ω] and \(S \in L^{\infty }(\hat{\mathbb{R}})\). Then

$$\displaystyle{ \forall f \in PW_{\varOmega },\quad f = T\sum _{n\in \mathbb{Z}}f(nT)\tau _{nT}s, }$$
(5.18)

where the convergence of (5.18) is in the \(L^{2}(\mathbb{R})\) norm and uniformly in \(\mathbb{R}\). One possible sampling function s is

$$\displaystyle{s(t) = \frac{\sin (2\pi \varOmega t)} {\pi t}.}$$

We can compute Fourier transforms numerically using the following result, whose proof, see [14], requires Theorem 17.

Theorem 18.

Let T,Ω > 0 satisfy the property that 2TΩ = 1, let N ≥ 2 be an even integer, and let \(f \in PW_{\varOmega } \cap L^{1}(\mathbb{R})\) . Consider the dilation f T (t) = Tf(Tt) as a continuous function on \(\mathbb{R}\) , as well as a function on \(\mathbb{Z}\) defined by m ↦ f T [m], where f T [m] = f T (m). Assume that \(f_{T} \in \ell^{1}(\mathbb{Z})\) . Then for every integer \(n \in [-\frac{N} {2}, \frac{N} {2} ]\) , we have

$$\displaystyle{ \hat{f}\left (\frac{2\varOmega n} {N} \right ) =\hat{ f}\left ( \frac{n} {NT}\right ) =\sum _{ m=0}^{N-1}(f_{ T})_{N}^{\circ }[m]W_{ N}^{mn}, }$$
(5.19)

where W N = e −2πi∕N and \((f_{T})_{N}^{\circ }[m] =\sum _{k\in \mathbb{Z}}f_{T}[m - kN]\) .

In practice, the computation (5.19) requires natural error estimates and the FFT.

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Benedetto, J.J., Begué, M.J. (2015). Fourier Operators in Applied Harmonic Analysis. In: Pfander, G. (eds) Sampling Theory, a Renaissance. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-19749-4_5

Download citation

Publish with us

Policies and ethics

Search

Navigation