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.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Notes
This implies that the number of cosets of \(H_n\) should be at most \(r^n\).
References
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)
Beckner, W.: Inequalities in Fourier analysis. Ann. Math. (2) 102(1), 159–182 (1975)
Bernal, A.: A note on the one-dimensional maximal function. Proc. R. Soc. Edinb. Sect. A 111(3–4), 325–328 (1989)
Bonami, A., Demange, B.: A survey on uncertainty principles related to quadratic forms. Collect. Math., (Vol. Extra):1–36 (2006)
Chistyakov, A.L.: On uncertainty relations for vector-valued operators. Teoret. Mat. Fiz. 27, 130–134 (1976)
Christensen, J.G.: The uncertainty principle for operators determined by Lie groups. J. Fourier Anal. Appl. 10(5), 541–544 (2004)
Cowling, M.G., Price, J.F.: Bandwidth versus time concentration: the Heisenberg-Pauli-Weyl inequality. SIAM J. Math. Anal. 15(1), 151–165 (1984)
Dall’Ara, G.M., Trevisan, D.: Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities. J. Geom. Anal. 25(4), 2262–2283 (2015)
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)
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)
Dembo, A., Cover, T.M., Thomas, J.A.: Information-theoretic inequalities. IEEE Trans. Inform. Theory 37(6), 1501–1518 (1991)
Donoho, D.L., Stark, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49(3), 906–931 (1989)
Fefferman, C.L.: The uncertainty principle. Bull. Am. Math. Soc. (N.S.) 9(2), 129–206 (1983)
Feng, T., Hollmann, H.D.L., Xiang, Q.: The shift bound for abelian codes and generalizations of the Donoho-Stark uncertainty principle. Preprint (2019)
Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3(3), 207–238 (1997)
Fournier, J.J.F.: Local complements to the Hausdorff-Young theorem. Mich. Math. J. 20, 263–276 (1973)
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)
Gilbert, J., Rzeszotnik, Z.: The norm of the Fourier transform on finite abelian groups. Ann. Inst. Fourier (Grenoble) 60(4), 1317–1346 (2010)
Grafakos, L.: Classical Fourier Analysis, vol. 249 of Graduate Texts in Mathematics. Springer, New York, third edition (2014)
Grafakos, L., Montgomery-Smith, S.: Best constants for uncentred maximal functions. Bull. London Math. Soc. 29(1), 60–64 (1997)
Hardy, G.H.: A theorem concerning Fourier transforms. J. London Math. Soc. 8, 227–231 (1933)
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)
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)
Hirschman Jr., I.I.: A note on entropy. Am. J. Math. 79, 152–156 (1957)
Hogan, J.A.: A qualitative uncertainty principle for unimodular groups of type \({\rm I}\). Trans. Am. Math. Soc. 340(2), 587–594 (1993)
Hörmander, L.: A uniqueness theorem of Beurling for Fourier transform pairs. Ark. Mat. 29(2), 237–240 (1991)
Jaming, P.: Nazarov’s uncertainty principles in higher dimension. J. Approx. Theory 149(1), 30–41 (2007)
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)
Jizba, P., Hayes, A., Dunningham, J.A.: New class of entropy-power-based uncertainty relations. J. Phys.: Conf. Ser. 880(1), 1–9 (2017)
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)
Kennard, E.H.: Zur Quantenmechanik einfacher Bewegungstypen. Z. Physik 44, 326–352 (1927)
Kovrijkine, O.: Some results related to the Logvinenko-Sereda theorem. Proc. Am. Math. Soc. 129(10), 3037–3047 (2001)
Kraus, K.: A further remark on uncertainty relations. Z. Physik 201, 134–141 (1967)
Lieb, E.H.: Gaussian kernels have only Gaussian maximizers. Invent. Math. 102(1), 179–208 (1990)
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)
Madiman, M., Ghassemi, F.: Combinatorial entropy power inequalities: a preliminary study of the Stam region. IEEE Trans. Inform. Theory 65(3), 1375–1386 (2019)
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)
Madiman, M., Melbourne, J., Xu, P.: Rogozin’s convolution inequality for locally compact groups. Preprint, arXiv:1705.00642 (2017)
Madiman, M., Wang, L., Woo, J.O.: Rényi entropy inequalities for sums in prime cyclic groups. Preprint, arXiv:1710.00812 (2017)
Madiman, M., Wang, L., Woo, J.O.: Majorization and Rényi entropy inequalities via Sperner theory. Discrete Math. 342(10), 2911–2923 (2019)
Martín, J., Milman, M.: Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces. Preprint, arXiv:1501.06556 (2015)
Matolcsi, T., Szűcs, J.: Intersection des mesures spectrales conjuguées. C. R. Acad. Sci. Paris Sér. A-B 277, A841–A843 (1973)
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)
Melas, A.D.: The best constant for the centered Hardy-Littlewood maximal inequality. Ann. Math. (2) 157(2), 647–688 (2003)
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)
Özaydin, M., Przebinda, T.: An entropy-based uncertainty principle for a locally compact abelian group. J. Funct. Anal. 215(1), 241–252 (2004)
Paneah, B.: Support dependent weighted norm estimates for Fourier transforms. J. Math. Anal. Appl. 189(2), 552–574 (1995)
Paneah, B.: Support-dependent weighted norm estimates for Fourier transforms. II. Duke Math. J. 92(2), 335–353 (1998)
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)
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)
Przebinda, T., DeBrunner, V., Özaydın, M.: The optimal transform for the discrete Hirschman uncertainty principle. IEEE Trans. Inform. Theory 47(5), 2086–2090 (2001)
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)
Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 573–574 (1929)
Rudin, W.: Fourier Analysis on Groups. Wiley Classics Library. Wiley, New York (1990). Reprint of the 1962 original, A Wiley-Interscience Publication
Sen, D.: The uncertainty relations in quantum mechanics. Curr. Sci. 107(2), 203–218 (2014)
Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(379–423), 623–656 (1948)
Smith, K.T.: The uncertainty principle on groups. SIAM J. Appl. Math. 50(3), 876–882 (1990)
Stam, A.J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inf. Control 2, 101–112 (1959)
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)
Stein, E.M., Strömberg, J.-O.: Behavior of maximal functions in \({ R}^{n}\) for large \(n\). Ark. Mat. 21(2), 259–269 (1983)
Tao, T.: An uncertainty principle for cyclic groups of prime order. Math. Res. Lett. 12(1), 121–127 (2005)
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
Wang, L., Madiman, M.: Beyond the entropy power inequality, via rearrangements. IEEE Trans. Inf. Theory 60(9), 5116–5137 (2014)
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
Zozor, S., Portesi, M., Vignat, C.: Some extensions of the uncertainty principle. Physica A 387(19–20), 4800–4808 (2008)
Zygmund, A.: Trigonometric Series, vol. I, II, 2nd edn. Cambridge University Press, New York (1959)
Acknowledgements
The authors are grateful to Philippe Jaming and to two anonymous referees for useful comments on an earlier draft of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hans G. Feichtinger.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported in part by the U.S. National Science Foundation through grants CCF-1346564 and DMS-1409504 (CAREER).
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00041-020-09737-7