Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.

MathSciNet review: 4076537
Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli
Title: Discrete harmonic analysis
Additional book information: Cambridge Studies in Advanced Mathematics, Vol. 172, Cambridge University Press, Cambridge, 2018, xiii+573 pp., ISBN 978-1-107-18233-2

References [Enhancements On Off] (What's this?)

  • Martin Aigner and Günter M. Ziegler, Proofs from The Book, 5th ed., Springer-Verlag, Berlin, 2014. Including illustrations by Karl H. Hofmann. MR 3288091, DOI 10.1007/978-3-662-44205-0
  • N. Alon, Eigenvalues and expanders, Combinatorica 6 (1986), no. 2, 83–96. Theory of computing (Singer Island, Fla., 1984). MR 875835, DOI 10.1007/BF02579166
  • N. Alon and V. D. Milman, $\lambda _1,$ isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B 38 (1985), no. 1, 73–88. MR 782626, DOI 10.1016/0095-8956(85)90092-9
  • Noga Alon, Oded Schwartz, and Asaf Shapira, An elementary construction of constant-degree expanders, Combin. Probab. Comput. 17 (2008), no. 3, 319–327. MR 2410389, DOI 10.1017/S0963548307008851
  • Noga Alon and Joel H. Spencer, The probabilistic method, 3rd ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2008. With an appendix on the life and work of Paul Erdős. MR 2437651, DOI 10.1002/9780470277331
  • L. Auslander and R. Tolimieri, Is computing with the finite Fourier transform pure or applied mathematics?, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 847–897. MR 546312, DOI 10.1090/S0273-0979-1979-14686-X
  • Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan’s property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. MR 2415834, DOI 10.1017/CBO9780511542749
  • N.N. Bogolyubov, On some ergodic properties of continuous transformation groups, Nauch. Zap. Kiev Univ. Phys.-Mat. Sb. 4:3 (1939), 45–53 (see also N.N. Bogolyubov, Seclected works, vol 1, Naukova Dumka, Kiev 1969. pp. 561–569).
  • Daniel Bump, Persi Diaconis, Angela Hicks, Laurent Miclo, and Harold Widom, An exercise(?) in Fourier analysis on the Heisenberg group, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 2, 263–288 (English, with English and French summaries). MR 3640891, DOI 10.5802/afst.1533
  • Peter Buser, Über eine Ungleichung von Cheeger, Math. Z. 158 (1978), no. 3, 245–252 (German). MR 478248, DOI 10.1007/BF01214795
  • Peter Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213–230. MR 683635
  • P. de lya Arp, R. I. Grigorchuk, and T. Chekerini-Sil′berstaĭn, Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces, Tr. Mat. Inst. Steklova 224 (1999), no. Algebra. Topol. Differ. Uravn. i ikh Prilozh., 68–111 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 1(224) (1999), 57–97. MR 1721355
  • Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli, Harmonic analysis on finite groups, Cambridge Studies in Advanced Mathematics, vol. 108, Cambridge University Press, Cambridge, 2008. Representation theory, Gelfand pairs and Markov chains. MR 2389056, DOI 10.1017/CBO9780511619823
  • Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli, Representation theory and harmonic analysis of wreath products of finite groups, London Mathematical Society Lecture Note Series, vol. 410, Cambridge University Press, Cambridge, 2014. MR 3202374
  • Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR 0402831
  • James W. Cooley, Lanczos and the FFT: a discovery before its time, Proceedings of the Cornelius Lanczos International Centenary Conference (Raleigh, NC, 1993) SIAM, Philadelphia, PA, 1994, pp. 3–9. MR 1298218
  • James W. Cooley and John W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comp. 19 (1965), 297–301. MR 178586, DOI 10.1090/S0025-5718-1965-0178586-1
  • G. C. Danielson and C. Lanczos, Some improvements in practical Fourier analysis and their application to X-ray scattering from liquids, J. Franklin Inst. 233 (1942), 365–380, 435–452. MR 6233, DOI 10.1016/S0016-0032(42)90767-1
  • H. Davenport and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. Reine Angew. Math. 172 (1935), 151–182 (German). MR 1581445, DOI 10.1515/crll.1935.172.151
  • Giuliana Davidoff, Peter Sarnak, and Alain Valette, Elementary number theory, group theory, and Ramanujan graphs, London Mathematical Society Student Texts, vol. 55, Cambridge University Press, Cambridge, 2003. MR 1989434, DOI 10.1017/CBO9780511615825
  • Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258
  • Persi Diaconis, Group representations in probability and statistics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 11, Institute of Mathematical Statistics, Hayward, CA, 1988. MR 964069
  • Jozef Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284 (1984), no. 2, 787–794. MR 743744, DOI 10.1090/S0002-9947-1984-0743744-X
  • Bernard Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960), 631–648. MR 140494, DOI 10.2307/2372974
  • P. Erdős, Über die Reihe $\sum \frac {1}{p}$, Mathematica, Zutphen B, 7 (1938), 1–2.
  • Shai Evra, Emmanuel Kowalski, and Alexander Lubotzky, Good cyclic codes and the uncertainty principle, Enseign. Math. 63 (2017), no. 3-4, 305–332. MR 3852174, DOI 10.4171/LEM/63-3/4-4
  • Joseph Fourier, Théorie analytique de la chaleur, Éditions Jacques Gabay, Paris, 1988 (French). Reprint of the 1822 original. MR 1414430
  • Daniel Goldstein, Robert M. Guralnick, and I. M. Isaacs, Inequalities for finite group permutation modules, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4017–4042. MR 2159698, DOI 10.1090/S0002-9947-05-03927-9
  • I. J. Good, The interaction algorithm and practical Fourier analysis, J. Roy. Statist. Soc. Ser. B 20 (1958), 361–372. MR 102888
  • Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), no. 2, 481–547. MR 2415379, DOI 10.4007/annals.2008.167.481
  • Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
  • R. I. Grigorchuk, Symmetrical random walks on discrete groups, Multicomponent random systems, Adv. Probab. Related Topics, vol. 6, Dekker, New York, 1980, pp. 285–325. MR 599539
  • Rostislav Grigorchuk and Pierre de la Harpe, Amenability and ergodic properties of topological groups: from Bogolyubov onwards, Groups, graphs and random walks, London Math. Soc. Lecture Note Ser., vol. 436, Cambridge Univ. Press, Cambridge, 2017, pp. 215–249. MR 3644011
  • R. I. Grigorchuk and A. M. Stëpin, Gibbs states on countable groups, Teor. Veroyatnost. i Primenen. 29 (1984), no. 2, 351–354 (Russian). MR 749922
  • Rostislav I. Grigorchuk and Andrzej Żuk, On the asymptotic spectrum of random walks on infinite families of graphs, Random walks and discrete potential theory (Cortona, 1997) Sympos. Math., XXXIX, Cambridge Univ. Press, Cambridge, 1999, pp. 188–204. MR 1802431
  • Misha Gromov and Larry Guth, Generalizations of the Kolmogorov-Barzdin embedding estimates, Duke Math. J. 161 (2012), no. 13, 2549–2603. MR 2988903, DOI 10.1215/00127094-1812840
  • Alexander Grothendieck, Formule de Lefschetz et rationalité des fonctions $L$, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 279, 41–55 (French). MR 1608788
  • Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
  • L. K. Hua and H. S. Vandiver, Characters over certain types of rings with applications to the theory of equations in a finite field, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 94–99. MR 28895, DOI 10.1073/pnas.35.2.94
  • A.N. Kolmogorov and Ya.M. Barzdin, On the realization of networks in three-dimensional space, in Selected Works of Kolmogorov, Volume 3, ed. Shiryaev, A. N., Kluwer Academic Publishers, Dordrecht, 1993.
  • Cornelius Lanczos, Applied analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1956. MR 0084175
  • W. C. Winnie Li, Number theory with applications, Series on University Mathematics, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1390759, DOI 10.1142/2716
  • Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0054173
  • Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, vol. 125, Birkhäuser Verlag, Basel, 1994. With an appendix by Jonathan D. Rogawski. MR 1308046, DOI 10.1007/978-3-0346-0332-4
  • Alexander Lubotzky, Expander graphs in pure and applied mathematics, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 1, 113–162. MR 2869010, DOI 10.1090/S0273-0979-2011-01359-3
  • A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), no. 3, 261–277. MR 963118, DOI 10.1007/BF02126799
  • James H. McClellan and Thomas W. Parks, Eigenvalue and eigenvector decomposition of the discrete Fourier transform, IEEE Trans. Audio Electroacoust. AU-20 (1972), no. 1, 66–74. MR 0399751
  • Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava, Interlacing families I: Bipartite Ramanujan graphs of all degrees, Ann. of Math. (2) 182 (2015), no. 1, 307–325. MR 3374962, DOI 10.4007/annals.2015.182.1.7
  • Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava, Interlacing families IV: Bipartite Ramanujan graphs of all sizes, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science—FOCS 2015, IEEE Computer Soc., Los Alamitos, CA, 2015, pp. 1358–1377. MR 3473375, DOI 10.1109/FOCS.2015.87
  • G. A. Margulis, Explicit constructions of expanders, Problemy Peredači Informacii 9 (1973), no. 4, 71–80 (Russian). MR 0484767
  • G. A. Margulis, Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators, Problemy Peredachi Informatsii 24 (1988), no. 1, 51–60 (Russian); English transl., Problems Inform. Transmission 24 (1988), no. 1, 39–46. MR 939574
  • David K. Maslen and Daniel N. Rockmore, The Cooley-Tukey FFT and group theory, Notices Amer. Math. Soc. 48 (2001), no. 10, 1151–1160. MR 1861656
  • R. Meshulam, private communication.
  • Melvyn B. Nathanson, Elementary methods in number theory, Graduate Texts in Mathematics, vol. 195, Springer-Verlag, New York, 2000. MR 1732941
  • J. von Neumann, Zur allgemeinen Theorie des Masses, Fund. Math. 13 (1929), 73–116 and 333 (= Collected works, vol. I, pp. 599–643).
  • A. Nilli, Tight estimates for eigenvalues of regular graphs, Electron. J. Combin. 11 (2004), no. 1, Note 9, 4. MR 2056091
  • Ilya Piatetski-Shapiro, Complex representations of $\textrm {GL}(2,\,K)$ for finite fields $K$, Contemporary Mathematics, vol. 16, American Mathematical Society, Providence, R.I., 1983. MR 696772
  • M.S. Pinkser, On the complexity of a concentrator, 7th International Teletraffic Conference (1973), 318/1–318/4.
  • L. Pontrjagin, Topological groups, Princeton University Press, Princeton, N.J., 1939 1958. Translated from the Russian by Emma Lehmer; (Fifth printing, 1958). MR 0090766
  • Ch.M. Rader, Discrete Fourier transforms when the number of data samples is prime, Proc. IEEE 56 (1968), 1107–1108.
  • Omer Reingold, Salil Vadhan, and Avi Wigderson, Entropy waves, the zig-zag graph product, and new constant-degree expanders, Ann. of Math. (2) 155 (2002), no. 1, 157–187. MR 1888797, DOI 10.2307/3062153
  • Donald J. Rose, Matrix identities of the fast Fourier transform, Linear Algebra Appl. 29 (1980), 423–443. MR 562772, DOI 10.1016/0024-3795(80)90253-0
  • Jean-Pierre Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke $T_p$, J. Amer. Math. Soc. 10 (1997), no. 1, 75–102 (French). MR 1396897, DOI 10.1090/S0894-0347-97-00220-8
  • Elias M. Stein and Rami Shakarchi, Fourier analysis, Princeton Lectures in Analysis, vol. 1, Princeton University Press, Princeton, NJ, 2003. An introduction. MR 1970295
  • Toshikazu Sunada, $L$-functions in geometry and some applications, Curvature and topology of Riemannian manifolds (Katata, 1985) Lecture Notes in Math., vol. 1201, Springer, Berlin, 1986, pp. 266–284. MR 859591, DOI 10.1007/BFb0075662
  • Terence Tao, An uncertainty principle for cyclic groups of prime order, Math. Res. Lett. 12 (2005), no. 1, 121–127. MR 2122735, DOI 10.4310/MRL.2005.v12.n1.a11
  • Terence Tao and Van Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. MR 2289012, DOI 10.1017/CBO9780511755149
  • A. Tarski, Sur les fonctions additives dans les classes abstraites et leurs applications au problème de la mesure, C.R. Séances Soc. Sci. Lettres Varsovie, Cl III 22 (1929), 114–117.
  • A. Tarski, Algebraische Fassung des Massproblems, Fund. Math. 31 (1938), 47–66.
  • Grzegorz Tomkowicz and Stan Wagon, The Banach-Tarski paradox, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 163, Cambridge University Press, New York, 2016. With a foreword by Jan Mycielski. MR 3616119
  • Charles Van Loan, Computational frameworks for the fast Fourier transform, Frontiers in Applied Mathematics, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1153025, DOI 10.1137/1.9781611970999
  • André Weil, L’intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 869, Hermann & Cie, Paris, 1940 (French). [This book has been republished by the author at Princeton, N. J., 1941.]. MR 0005741
  • André Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497–508. MR 29393, DOI 10.1090/S0002-9904-1949-09219-4
  • S. Winograd, On computing the discrete Fourier transform, Math. Comp. 32 (1978), no. 141, 175–199. MR 468306, DOI 10.1090/S0025-5718-1978-0468306-4
  • Joseph A. Wolf, Harmonic analysis on commutative spaces, Mathematical Surveys and Monographs, vol. 142, American Mathematical Society, Providence, RI, 2007. MR 2328043, DOI 10.1090/surv/142

  • Review Information:

    Reviewer: Rostislav Grigorchuk
    Affiliation: Department of Mathematics,Texas A & M University, College Station, Texas
    Journal: Bull. Amer. Math. Soc. 57 (2020), 325-337
    Published electronically: July 29, 2019
    Review copyright: © Copyright 2019 American Mathematical Society