The norm of the Fourier transform on finite abelian groups
Annales de l'Institut Fourier, Volume 60 (2010) no. 4, p. 1317-1346
For 1p,q we calculate the norm of the Fourier transform from the L p space on a finite abelian group to the L q space on the dual group.
Pour les valeurs de p et q comprises entre 1 et l’infini, nous déterminons la norme de la transformée de Fourier de l’espace L p d’un groupe abélien fini vers l’espace L q du groupe dual.
DOI : https://doi.org/10.5802/aif.2556
Classification:  42C40,  43A15,  43A25
Keywords: Fourier transform, finite abelian groups, wave packets, biunimodular functions
@article{AIF_2010__60_4_1317_0,
     author = {Gilbert, John and Rzeszotnik, Ziemowit},
     title = {The norm of the Fourier transform on finite abelian groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {4},
     year = {2010},
     pages = {1317-1346},
     doi = {10.5802/aif.2556},
     zbl = {1202.42065},
     mrnumber = {2722243},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2010__60_4_1317_0}
}
Gilbert, John; Rzeszotnik, Ziemowit. The norm of the Fourier transform on finite abelian groups. Annales de l'Institut Fourier, Volume 60 (2010) no. 4, pp. 1317-1346. doi : 10.5802/aif.2556. https://aif.centre-mersenne.org/item/AIF_2010__60_4_1317_0/

[1] Adams, C. M. Constructing symmetric ciphers using the CAST design procedure, Des. Codes Cryptography, Tome 12 (1997) no. 3, pp. 283-316 | Article | MR 1473036 | Zbl 0880.94011

[2] Adams, C. M.; Tavares, S. E. Generating bent sequences, Discrete Appl. Math., Tome 39 (1992) no. 2, pp. 155-159 | Article | MR 1184686 | Zbl 0767.94004

[3] Agievich, S. On the representation of bent functions by bent rectangles, Probabilistic Methods in Discrete Mathematics: Proceedings of the Fifth International Petrozavodsk Conference, Utrecht, Boston: VSP (2002), pp. 121-135

[4] Babenko, K. I. An inequality in the theory of Fourier integrals, Izv. Akad. Nauk SSSR Ser. Mat., Tome 25 (1961), pp. 531-542 | MR 138939 | Zbl 0122.34404

[5] Beckner, W. Inequalities in Fourier analysis, Ann. Math. (2), Tome 102 (1975), pp. 159-182 | Article | MR 385456 | Zbl 0338.42017

[6] Björck, G. Functions of modulus 1 on Z n whose Fourier transforms have constant modulus, and “cyclic n-roots”, Recent advances in Fourier analysis and its applications, Proc. NATO/ASI, Il Ciocco/Italy 1989, NATO ASI Ser., Ser. C 315 (1990) | MR 1081347 | Zbl 0726.43004

[7] Björck, G.; Fröberg, R. A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic n-roots, J. Symb. Comput., Tome 12 (1991) no. 3, pp. 329-336 | Article | MR 1128248 | Zbl 0751.12001

[8] Björck, G.; Saffari, B. New classes of finite unimodular sequences with unimodular Fourier transforms. Circulant Hadamard matrices with complex entries, C. R. Acad. Sci., Paris, Sér. I, Tome 320 (1995) no. 3, pp. 319-324 | MR 1320378 | Zbl 0846.11016

[9] Byrnes, J. S. On polynomials with coefficients of modulus one, Bull. Lond. Math. Soc., Tome 9 (1997), pp. 171-176 | Article | MR 486435 | Zbl 0364.30004

[10] Carlet, C. Two new classes of bent functions, Helleseth, Tor (ed.), Advances in cryptology - EUROCRYPT ’93. Lect. Notes Comput. Sci. 765, Springer, Berlin (1994) | MR 1290331 | Zbl 0951.94542

[11] Casazza, P. G.; Fickus, M. Chirps on finite cyclic groups, Proc. SPIE, Tome 5914 (2005), pp. 175-180

[12] Chang, D. K. Binary bent sequences of order 64, Util. Math., Tome 52 (1997), pp. 141-151 | MR 1605743 | Zbl 0926.94019

[13] Coifman, R. R.; Meyer, Y.; Wickerhauser, M. V.; Ruskai, M. B.; Al. Wavelet analysis and signal processing, Wavelets and their applications, Jones and Bartlett Publishers, Boston, MA (1992), pp. 153-178 | MR 1187341 | Zbl 0792.94004

[14] Coifman, R. R.; Wickerhauser, M. V. Entropy-based algorithms for best basis selection, IEEE Trans. Inf. Theory, Tome 38 (1992) no. 2, Pt. 2, pp. 713-718 | Article | Zbl 0849.94005

[15] Dillon, J. F. Elementary Hadamard difference sets, Proc. 6th Southeast. Conf. Comb., Graph Theor., and Comput.; Boca Raton, Fl (1975) | MR 409221 | Zbl 0346.05003

[16] Dobbertin, H.; Leander, G. Cryptographer’s Toolkit for Construction of 8-Bit Bent Functions, Cryptology ePrint Archive, Report 2005/089 (2005)

[17] Donoho, D. L.; Stark, P. B. Uncertainty principles and signal recovery, SIAM J. Appl. Math., Tome 49 (1989) no. 3, pp. 906-931 | Article | MR 997928 | Zbl 0689.42001

[18] Fefferman, C. Pointwise convergence of Fourier series, Ann. Math. (2), Tome 98 (1973), pp. 551-571 | Article | MR 340926 | Zbl 0268.42009

[19] Feichtinger, H. G.; Hazewinkel, M.; Kaiblinger, N.; Matusiak, E.; Neuhauser, M. Metaplectic operators on C n , preprint | Zbl 1142.22007

[20] Haagerup, U. Orthogonal maximal abelian *-subalgebras of the n×n matrices and cyclic n-roots., Doplicher, S. (ed.) et al., Operator algebras and quantum field theory. Accademia Nazionale dei Lincei, Roma, Italy. Cambridge, MA: International Press (1997) | MR 1491124 | Zbl 0914.46045

[21] Hardy, G. H.; Littlewood, J. E. Some new properties of Fourier constants, Math. Ann., Tome 97 (1927), pp. 159-209 | Article | MR 1512359

[22] Herley, C.; Xiong, Z.; Ramchandran, K.; Orchard, M. T. Joint space-frequency segmentation using balanced wavelet packet trees for least-cost image representation, IEEE Trans. on Image Proc., Tome 6 (1997) no. 9, pp. 1213-1230 | Article

[23] Hewitt, E.; Hirschman, I. A maximum problem in harmonic analysis, Am. J. Math., Tome 76 (1954), pp. 839-852 | Article | MR 65034 | Zbl 0056.10504

[24] Hewitt, E.; Ross, K. A. Abstract harmonic analysis., Berlin-Heidelberg-New York: Springer-Verlag VIII Tome 2 (1970) | MR 262773 | Zbl 0115.10603

[25] Huang, Y.; Pollak, I.; Bouman, C. A. Image Compression with Multitree Tilings, Proc. ICASSP-2005, Philadelphia, PA. (March 2005)

[26] Lacey, M.; Thiele, C. L p estimates on the bilinear Hilbert transform for 2<p<, Ann. Math. (2), Tome 146 (1997) no. 3, pp. 693-724 | Article | MR 1491450 | Zbl 0914.46034

[27] Lacey, M.; Thiele, C. On Calderón’s conjecture, Ann. Math. (2), Tome 149 (1999) no. 2, pp. 475-496 | Article | MR 1689336 | Zbl 0934.42012

[28] Leung, Ka Hin; Ma, Siu Lun; Schmidt, B. Nonexistence of abelian difference sets: Lander’s conjecture for prime power orders, Trans. Am. Math. Soc., Tome 356 (2004) no. 11, pp. 4343-4358 | Article | MR 2067122 | Zbl 1043.05025

[29] Li, T. Y.; Li, Xing Finding mixed cells in the mixed volume computation, Found. Comput. Math., Tome 1 (2001) no. 2, pp. 161-181 | Article | MR 1830034 | Zbl 1012.65019

[30] Lieb, E. H. Gaussian kernels have only Gaussian maximizers, Invent. Math., Tome 102 (1990) no. 1, pp. 179-208 | Article | MR 1069246 | Zbl 0726.42005

[31] Littlewood, J. E. On the mean values of certain trigonometrical polynomials. II, Ill. J. Math., Tome 6 (1962), pp. 1-39 | MR 141935 | Zbl 0108.05802

[32] Macwilliams, F. J.; Sloane, N. J. A. The theory of error-correcting codes, North-Holland Mathematical Library, Amsterdam - New York - Oxford Tome 16 (1977) | Zbl 0369.94008

[33] Matusiak, E.; Özaydin, M.; Przebinda, T. The Donoho–Stark uncertainty principle for a finite abelian group, Acta Math. Univ. Comen. New Ser., Tome 73 (2004) no. 2, pp. 155-160 | MR 2122203 | Zbl 1100.43003

[34] Preneel, B.; Van Leekwijck, W.; Van Linden, L.; Govaerts, R.; Vandewalle, J. Propagation characteristics of Boolean functions, Advances in Cryptology, Proc. Workshop, EUROCRYPT ’90, Lect. Notes Comput. Sci. 473 (1991) | MR 1102479 | Zbl 0764.94024

[35] Richman, M. S.; Parks, T. W.; Shenoy, R. G. Discrete-time, discrete-frequency, time-frequency analysis, IEEE Trans. Signal Process., Tome 46 (1998) no. 6, pp. 1517-1527 | Article | Zbl 1010.94526

[36] Rothaus, O. S. On “bent” functions., J. Comb. Theory, Ser. A, Tome 20 (1976), pp. 300-305 | Article | MR 403988 | Zbl 0336.12012

[37] Ryser, H. J. Combinatorial mathematics, John Wiley and Sons, New York (1963) | MR 150048 | Zbl 0112.24806

[38] Saffari, B. Some polynomial extremal problems which emerged in the twentieth century, Byrnes, James S. (ed.), Twentieth century harmonic analysis–a celebration. Proceedings of the NATO Advanced Study Institute, Il Ciocco, Italy, July 2-15, 2000. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 33 (2001) | MR 1858787 | Zbl 0996.42001

[39] Schmidt, B. Cyclotomic integers and finite geometry, J. Am. Math. Soc., Tome 12 (1999) no. 4, pp. 929-952 | Article | MR 1671453 | Zbl 0939.05016

[40] Schmidt, B. Towards Ryser’s conjecture, Casacuberta, Carles (ed.) et al., 3rd European congress of mathematics (ECM), Barcelona, Spain, July 10-14, 2000. Volume I. Basel: Birkhäuser. Prog. Math. 201 (2001) | MR 1905341 | Zbl 1030.05018

[41] Takeda, A.; Kojima, M.; Fujisawa, K. Enumeration of all solutions of a combinatorial linear inequality system arising from the polyhedral homotopy continuation method, J. Oper. Res. Soc. Japan, Tome 45 (2002) no. 1, pp. 64-82 | MR 1898623 | Zbl 1031.65074

[42] Tao, T. An uncertainty principle for cyclic groups of prime order, Math. Res. Lett., Tome 12 (2005) no. 1, pp. 121-127 | MR 2122735 | Zbl 1080.42002

[43] Thiele, C.; Villemoes, L. F. A fast algorithm for adapted time-frequency tilings, Appl. Comput. Harmon. Anal., Tome 3 (1996) no. 2, pp. 91-99 | Article | MR 1385046 | Zbl 0857.65148

[44] Turyn, R. J. Character sums and difference sets, Pac. J. Math., Tome 15 (1965), pp. 319-346 | MR 179098 | Zbl 0135.05403

[45] Xia, X. G. Discrete chirp-Fourier transform and its application in chirp rate estimation, IEEE Trans. on Signal Processing, Tome 48(11) (2000), pp. 3122-3133 | MR 1791082 | Zbl 0979.94024

[46] Yarlagadda, R.; Hershey, J. E. Analysis and synthesis of bent sequences, Proc. IEE, Tome 136, Pt. E. (1989), pp. 112-123