Pour les valeurs de et comprises entre et l’infini, nous déterminons la norme de la transformée de Fourier de l’espace d’un groupe abélien fini vers l’espace du groupe dual.
For we calculate the norm of the Fourier transform from the space on a finite abelian group to the space on the dual group.
@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}, volume = {60}, year = {2010}, pages = {1317-1346}, doi = {10.5802/aif.2556}, zbl = {1202.42065}, mrnumber = {2722243}, language = {en}, url = {http://dml.mathdoc.fr/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, Tome 60 (2010) pp. 1317-1346. doi : 10.5802/aif.2556. http://gdmltest.u-ga.fr/item/AIF_2010__60_4_1317_0/
[1] 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] Generating bent sequences, Discrete Appl. Math., Tome 39 (1992) no. 2, pp. 155-159 | Article | MR 1184686 | Zbl 0767.94004
[3] 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] 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] Inequalities in Fourier analysis, Ann. Math. (2), Tome 102 (1975), pp. 159-182 | Article | MR 385456 | Zbl 0338.42017
[6] Functions of modulus 1 on 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] A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic -roots, J. Symb. Comput., Tome 12 (1991) no. 3, pp. 329-336 | Article | MR 1128248 | Zbl 0751.12001
[8] 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] On polynomials with coefficients of modulus one, Bull. Lond. Math. Soc., Tome 9 (1997), pp. 171-176 | Article | MR 486435 | Zbl 0364.30004
[10] 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] Chirps on finite cyclic groups, Proc. SPIE, Tome 5914 (2005), pp. 175-180
[12] Binary bent sequences of order 64, Util. Math., Tome 52 (1997), pp. 141-151 | MR 1605743 | Zbl 0926.94019
[13] 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] 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] Elementary Hadamard difference sets, Proc. 6th Southeast. Conf. Comb., Graph Theor., and Comput.; Boca Raton, Fl (1975) | MR 409221 | Zbl 0346.05003
[16] Cryptographer’s Toolkit for Construction of -Bit Bent Functions, Cryptology ePrint Archive, Report 2005/089 (2005)
[17] Uncertainty principles and signal recovery, SIAM J. Appl. Math., Tome 49 (1989) no. 3, pp. 906-931 | Article | MR 997928 | Zbl 0689.42001
[18] Pointwise convergence of Fourier series, Ann. Math. (2), Tome 98 (1973), pp. 551-571 | Article | MR 340926 | Zbl 0268.42009
[19] Metaplectic operators on , preprint | Zbl 1142.22007
[20] Orthogonal maximal abelian -subalgebras of the matrices and cyclic -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] Some new properties of Fourier constants, Math. Ann., Tome 97 (1927), pp. 159-209 | Article | MR 1512359
[22] 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] A maximum problem in harmonic analysis, Am. J. Math., Tome 76 (1954), pp. 839-852 | Article | MR 65034 | Zbl 0056.10504
[24] Abstract harmonic analysis., Berlin-Heidelberg-New York: Springer-Verlag VIII Tome 2 (1970) | MR 262773 | Zbl 0115.10603
[25] Image Compression with Multitree Tilings, Proc. ICASSP-2005, Philadelphia, PA. (March 2005)
[26] estimates on the bilinear Hilbert transform for , Ann. Math. (2), Tome 146 (1997) no. 3, pp. 693-724 | Article | MR 1491450 | Zbl 0914.46034
[27] On Calderón’s conjecture, Ann. Math. (2), Tome 149 (1999) no. 2, pp. 475-496 | Article | MR 1689336 | Zbl 0934.42012
[28] 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] 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] Gaussian kernels have only Gaussian maximizers, Invent. Math., Tome 102 (1990) no. 1, pp. 179-208 | Article | MR 1069246 | Zbl 0726.42005
[31] On the mean values of certain trigonometrical polynomials. II, Ill. J. Math., Tome 6 (1962), pp. 1-39 | MR 141935 | Zbl 0108.05802
[32] The theory of error-correcting codes, North-Holland Mathematical Library, Amsterdam - New York - Oxford Tome 16 (1977) | Zbl 0369.94008
[33] 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] Propagation characteristics of Boolean functions, Advances in Cryptology, Proc. Workshop, EUROCRYPT ’90, Lect. Notes Comput. Sci. 473 (1991) | MR 1102479 | Zbl 0764.94024
[35] Discrete-time, discrete-frequency, time-frequency analysis, IEEE Trans. Signal Process., Tome 46 (1998) no. 6, pp. 1517-1527 | Article | Zbl 1010.94526
[36] On “bent” functions., J. Comb. Theory, Ser. A, Tome 20 (1976), pp. 300-305 | Article | MR 403988 | Zbl 0336.12012
[37] Combinatorial mathematics, John Wiley and Sons, New York (1963) | MR 150048 | Zbl 0112.24806
[38] 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] Cyclotomic integers and finite geometry, J. Am. Math. Soc., Tome 12 (1999) no. 4, pp. 929-952 | Article | MR 1671453 | Zbl 0939.05016
[40] 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] 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] 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] 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] Character sums and difference sets, Pac. J. Math., Tome 15 (1965), pp. 319-346 | MR 179098 | Zbl 0135.05403
[45] 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] Analysis and synthesis of bent sequences, Proc. IEE, Tome 136, Pt. E. (1989), pp. 112-123