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The Norm of the Fourier Transform on Compact or Discrete Abelian Groups

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Abstract

We calculate the norm of the Fourier operator from \(L^p(X)\) to \(L^q({\hat{X}})\) when X is an infinite locally compact abelian group that is, furthermore, compact or discrete. This subsumes the sharp Hausdorff–Young inequality on such groups. In particular, we identify the region in (p, q)-space where the norm is infinite, generalizing a result of Fournier, and setting up a contrast with the case of finite abelian groups, where the norm was determined by Gilbert and Rzeszotnik. As an application, uncertainty principles on such groups expressed in terms of Rényi entropies are discussed.

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Notes

  1. This implies that the number of cosets of \(H_n\) should be at most \(r^n\).

References

  1. Amrein, W.O., Berthier, A.M.: On support properties of \(L^{p}\)-functions and their Fourier transforms. J. Funct. Anal. 24(3), 258–267 (1977)

    MATH  Google Scholar 

  2. Beckner, W.: Inequalities in Fourier analysis. Ann. Math. (2) 102(1), 159–182 (1975)

    MathSciNet  MATH  Google Scholar 

  3. Bernal, A.: A note on the one-dimensional maximal function. Proc. R. Soc. Edinb. Sect. A 111(3–4), 325–328 (1989)

    MathSciNet  MATH  Google Scholar 

  4. Bonami, A., Demange, B.: A survey on uncertainty principles related to quadratic forms. Collect. Math., (Vol. Extra):1–36 (2006)

  5. Chistyakov, A.L.: On uncertainty relations for vector-valued operators. Teoret. Mat. Fiz. 27, 130–134 (1976)

    MathSciNet  Google Scholar 

  6. Christensen, J.G.: The uncertainty principle for operators determined by Lie groups. J. Fourier Anal. Appl. 10(5), 541–544 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Cowling, M.G., Price, J.F.: Bandwidth versus time concentration: the Heisenberg-Pauli-Weyl inequality. SIAM J. Math. Anal. 15(1), 151–165 (1984)

    MathSciNet  Google Scholar 

  8. Dall’Ara, G.M., Trevisan, D.: Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities. J. Geom. Anal. 25(4), 2262–2283 (2015)

    MathSciNet  MATH  Google Scholar 

  9. de Bruijn, N. G.: Uncertainty principles in Fourier analysis. In Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965), pages 57–71. Academic Press, New York (1967)

  10. DeBrunner, V., Havlicek, J.P., Przebinda, T., Özaydın, M.: Entropy-based uncertainty measures for \(L^2(\mathbb{R}^n)\), \(l^2(\mathbb{Z})\), and \(l^2(\mathbb{Z}/N\mathbb{Z})\) with a Hirschman optimal transform for \(l^2(\mathbb{Z}/N\mathbb{Z})\). IEEE Trans. Signal Process. 53(8, part 1), 2690–2699 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Dembo, A., Cover, T.M., Thomas, J.A.: Information-theoretic inequalities. IEEE Trans. Inform. Theory 37(6), 1501–1518 (1991)

    MathSciNet  MATH  Google Scholar 

  12. Donoho, D.L., Stark, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49(3), 906–931 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Fefferman, C.L.: The uncertainty principle. Bull. Am. Math. Soc. (N.S.) 9(2), 129–206 (1983)

    MathSciNet  MATH  Google Scholar 

  14. Feng, T., Hollmann, H.D.L., Xiang, Q.: The shift bound for abelian codes and generalizations of the Donoho-Stark uncertainty principle. Preprint (2019)

  15. Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3(3), 207–238 (1997)

    MathSciNet  MATH  Google Scholar 

  16. Fournier, J.J.F.: Local complements to the Hausdorff-Young theorem. Mich. Math. J. 20, 263–276 (1973)

    MathSciNet  MATH  Google Scholar 

  17. Garcia, S. R., Karaali, G., Katz, D. J.: On Chebotarëv’s nonvanishing minors theorem and the Biró-Meshulam-Tao discrete uncertainty principle. Preprint, arXiv:1807.07648v2, (2019)

  18. Gilbert, J., Rzeszotnik, Z.: The norm of the Fourier transform on finite abelian groups. Ann. Inst. Fourier (Grenoble) 60(4), 1317–1346 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Grafakos, L.: Classical Fourier Analysis, vol. 249 of Graduate Texts in Mathematics. Springer, New York, third edition (2014)

  20. Grafakos, L., Montgomery-Smith, S.: Best constants for uncentred maximal functions. Bull. London Math. Soc. 29(1), 60–64 (1997)

    MathSciNet  MATH  Google Scholar 

  21. Hardy, G.H.: A theorem concerning Fourier transforms. J. London Math. Soc. 8, 227–231 (1933)

    MathSciNet  MATH  Google Scholar 

  22. Havin, V., Jöricke, B.: The Uncertainty Principle in Harmonic Analysis, vol. 28 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer, Berlin (1994)

  23. Havin, V.P., Jöricke, B.: The uncertainty principle in harmonic analysis [ MR1129019 (93e:42001)]. In Commutative harmonic analysis, III, volume 72 of Encyclopaedia Math. Sci., pages 177–259, 261–266. Springer, Berlin (1995)

  24. Hirschman Jr., I.I.: A note on entropy. Am. J. Math. 79, 152–156 (1957)

    MathSciNet  MATH  Google Scholar 

  25. Hogan, J.A.: A qualitative uncertainty principle for unimodular groups of type \({\rm I}\). Trans. Am. Math. Soc. 340(2), 587–594 (1993)

    MathSciNet  MATH  Google Scholar 

  26. Hörmander, L.: A uniqueness theorem of Beurling for Fourier transform pairs. Ark. Mat. 29(2), 237–240 (1991)

    MathSciNet  MATH  Google Scholar 

  27. Jaming, P.: Nazarov’s uncertainty principles in higher dimension. J. Approx. Theory 149(1), 30–41 (2007)

    MathSciNet  MATH  Google Scholar 

  28. Jog, V., Anantharam, V.: The entropy power inequality and Mrs. Gerber’s lemma for groups of order \(2^n\). IEEE Trans. Inform. Theory 60(7), 3773–3786 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Jizba, P., Hayes, A., Dunningham, J.A.: New class of entropy-power-based uncertainty relations. J. Phys.: Conf. Ser. 880(1), 1–9 (2017)

    Google Scholar 

  30. Kaniuth, E.: Minimizing functions for an uncertainty principle on locally compact groups of bounded representation dimension. Proc. Am. Math. Soc. 135(1), 217–227 (2007)

    MathSciNet  MATH  Google Scholar 

  31. Kennard, E.H.: Zur Quantenmechanik einfacher Bewegungstypen. Z. Physik 44, 326–352 (1927)

    MATH  Google Scholar 

  32. Kovrijkine, O.: Some results related to the Logvinenko-Sereda theorem. Proc. Am. Math. Soc. 129(10), 3037–3047 (2001)

    MathSciNet  MATH  Google Scholar 

  33. Kraus, K.: A further remark on uncertainty relations. Z. Physik 201, 134–141 (1967)

    MathSciNet  Google Scholar 

  34. Lieb, E.H.: Gaussian kernels have only Gaussian maximizers. Invent. Math. 102(1), 179–208 (1990)

    MathSciNet  MATH  Google Scholar 

  35. Logvinenko, V.N., Sereda, Ju.F.: Equivalent norms in spaces of entire functions of exponential type. Teor. Funkciĭ Funkcional. Anal. i Priložen., (Vyp. 20):102–111, 175 (1974)

  36. Madiman, M., Ghassemi, F.: Combinatorial entropy power inequalities: a preliminary study of the Stam region. IEEE Trans. Inform. Theory 65(3), 1375–1386 (2019)

    MathSciNet  MATH  Google Scholar 

  37. Madiman, M., Melbourne, J., Xu, P.: Forward and reverse entropy power inequalities in convex geometry. In: E. Carlen, M. Madiman, E. M. Werner, (eds), Convexity and Concentration, volume 161 of IMA Volumes in Mathematics and its Applications, pages 427–485. Springer (2017)

  38. Madiman, M., Melbourne, J., Xu, P.: Rogozin’s convolution inequality for locally compact groups. Preprint, arXiv:1705.00642 (2017)

  39. Madiman, M., Wang, L., Woo, J.O.: Rényi entropy inequalities for sums in prime cyclic groups. Preprint, arXiv:1710.00812 (2017)

  40. Madiman, M., Wang, L., Woo, J.O.: Majorization and Rényi entropy inequalities via Sperner theory. Discrete Math. 342(10), 2911–2923 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Martín, J., Milman, M.: Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces. Preprint, arXiv:1501.06556 (2015)

  42. Matolcsi, T., Szűcs, J.: Intersection des mesures spectrales conjuguées. C. R. Acad. Sci. Paris Sér. A-B 277, A841–A843 (1973)

    MATH  Google Scholar 

  43. Matusiak, E., Özaydın, M., Przebinda, T.: The Donoho-Stark uncertainty principle for a finite abelian group. Acta Math. Univ. Comenian. (N.S.) 73(2), 155–160 (2004)

    MathSciNet  MATH  Google Scholar 

  44. Melas, A.D.: The best constant for the centered Hardy-Littlewood maximal inequality. Ann. Math. (2) 157(2), 647–688 (2003)

    MathSciNet  MATH  Google Scholar 

  45. Nazarov, F.L.: On the theorems of Turán, Amrein and Berthier, and Zygmund. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 201(Issled. po Linein. Oper. Teor. Funktsii. 20):117–123, 191, (1992)

  46. Özaydin, M., Przebinda, T.: An entropy-based uncertainty principle for a locally compact abelian group. J. Funct. Anal. 215(1), 241–252 (2004)

    MathSciNet  MATH  Google Scholar 

  47. Paneah, B.: Support dependent weighted norm estimates for Fourier transforms. J. Math. Anal. Appl. 189(2), 552–574 (1995)

    MathSciNet  MATH  Google Scholar 

  48. Paneah, B.: Support-dependent weighted norm estimates for Fourier transforms. II. Duke Math. J. 92(2), 335–353 (1998)

    MathSciNet  MATH  Google Scholar 

  49. Parui, S., Thangavelu, S.: Variations on a theorem of Cowling and Price with applications to nilpotent Lie groups. J. Aust. Math. Soc. 82(1), 11–27 (2007)

    MathSciNet  MATH  Google Scholar 

  50. Przebinda, T.: Three uncertainty principles for an abelian locally compact group. In Representations of real and \(p\)-adic groups, volume 2 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., pages 1–18. Singapore University Press, Singapore (2004)

  51. Przebinda, T., DeBrunner, V., Özaydın, M.: The optimal transform for the discrete Hirschman uncertainty principle. IEEE Trans. Inform. Theory 47(5), 2086–2090 (2001)

    MathSciNet  MATH  Google Scholar 

  52. Ram Murty, M., Whang, J.P.: The uncertainty principle and a generalization of a theorem of Tao. Linear Algebra Appl. 437(1), 214–220 (2012)

    MathSciNet  MATH  Google Scholar 

  53. Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 573–574 (1929)

    Google Scholar 

  54. Rudin, W.: Fourier Analysis on Groups. Wiley Classics Library. Wiley, New York (1990). Reprint of the 1962 original, A Wiley-Interscience Publication

  55. Sen, D.: The uncertainty relations in quantum mechanics. Curr. Sci. 107(2), 203–218 (2014)

    Google Scholar 

  56. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(379–423), 623–656 (1948)

    MathSciNet  MATH  Google Scholar 

  57. Smith, K.T.: The uncertainty principle on groups. SIAM J. Appl. Math. 50(3), 876–882 (1990)

    MathSciNet  Google Scholar 

  58. Stam, A.J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inf. Control 2, 101–112 (1959)

    MathSciNet  MATH  Google Scholar 

  59. Stein, E.M.: The development of square functions in the work of A. Zygmund. Bull. Am. Math. Soc. (N.S.) 7(2), 359–376 (1982)

    MathSciNet  MATH  Google Scholar 

  60. Stein, E.M., Strömberg, J.-O.: Behavior of maximal functions in \({ R}^{n}\) for large \(n\). Ark. Mat. 21(2), 259–269 (1983)

    MathSciNet  MATH  Google Scholar 

  61. Tao, T.: An uncertainty principle for cyclic groups of prime order. Math. Res. Lett. 12(1), 121–127 (2005)

    MathSciNet  MATH  Google Scholar 

  62. Thangavelu, S., An Introduction to the Uncertainty Principle, vol. 217 Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, (2004). Hardy’s theorem on Lie groups. With a foreword by Gerald B. Folland

  63. Wang, L., Madiman, M.: Beyond the entropy power inequality, via rearrangements. IEEE Trans. Inf. Theory 60(9), 5116–5137 (2014)

    MathSciNet  MATH  Google Scholar 

  64. Weyl, H.: The Theory of Groups and Quantum Mechanics. Dover Publications, Inc., New York, (1950). Translated from the second (revised) German edition by H. P. Roberton, Reprint of the 1931 English translation

  65. Zozor, S., Portesi, M., Vignat, C.: Some extensions of the uncertainty principle. Physica A 387(19–20), 4800–4808 (2008)

    MathSciNet  Google Scholar 

  66. Zygmund, A.: Trigonometric Series, vol. I, II, 2nd edn. Cambridge University Press, New York (1959)

    MATH  Google Scholar 

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Acknowledgements

The authors are grateful to Philippe Jaming and to two anonymous referees for useful comments on an earlier draft of this paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    Mokshay Madiman

  2. Department of Mathematics and Statistics, Eastern Michigan University, Ypsilanti, MI, USA

    Peng Xu

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  1. Mokshay Madiman
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Correspondence to Mokshay Madiman.

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This work was supported in part by the U.S. National Science Foundation through grants CCF-1346564 and DMS-1409504 (CAREER).

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Madiman, M., Xu, P. The Norm of the Fourier Transform on Compact or Discrete Abelian Groups. J Fourier Anal Appl 26, 37 (2020). https://doi.org/10.1007/s00041-020-09737-7

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