Une nouvelle propriété des suites de Rudin-Shapiro
Annales de l'Institut Fourier, Tome 37 (1987) no. 2, pp. 115-138.

Les suites de Rudin-Shapiro ont des propriétés extrémales en analyse harmonique. En remarquant qu’une telle suite est reconnaissable par un automate fini, nous en décrivons explicitement le spectre (type spectral maximal, multiplicité spectrale fonction multiplicité). Nous établissons par exemple, que la suite de Rudin-Shapiro généralisée à l’ordre q contient dans son spectre une composante de Lebesgue, de multiplicité qϕ(q).

The Rudin-Shapiro sequences have extremal properties in harmonic analysis. Using the fact that such a sequence is an automaton-sequence, we describe explicitely its spectrum (maximal spectral type, spectral multiplicity, multiplicity function). For example, we prove that the q-generalized Rudin-Shapiro sequence contains in its spectrum a Lebesgue-component, with multiplicity equal to qϕ(q).

@article{AIF_1987__37_2_115_0,
     author = {Queffelec, Martine},
     title = {Une nouvelle propri\'et\'e des suites de {Rudin-Shapiro}},
     journal = {Annales de l'Institut Fourier},
     pages = {115--138},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {37},
     number = {2},
     year = {1987},
     doi = {10.5802/aif.1089},
     mrnumber = {88m:11060},
     zbl = {0597.10054},
     language = {fr},
     url = {https://www.numdam.org/articles/10.5802/aif.1089/}
}
TY  - JOUR
AU  - Queffelec, Martine
TI  - Une nouvelle propriété des suites de Rudin-Shapiro
JO  - Annales de l'Institut Fourier
PY  - 1987
SP  - 115
EP  - 138
VL  - 37
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - https://www.numdam.org/articles/10.5802/aif.1089/
DO  - 10.5802/aif.1089
LA  - fr
ID  - AIF_1987__37_2_115_0
ER  - 
%0 Journal Article
%A Queffelec, Martine
%T Une nouvelle propriété des suites de Rudin-Shapiro
%J Annales de l'Institut Fourier
%D 1987
%P 115-138
%V 37
%N 2
%I Institut Fourier
%C Grenoble
%U https://www.numdam.org/articles/10.5802/aif.1089/
%R 10.5802/aif.1089
%G fr
%F AIF_1987__37_2_115_0
Queffelec, Martine. Une nouvelle propriété des suites de Rudin-Shapiro. Annales de l'Institut Fourier, Tome 37 (1987) no. 2, pp. 115-138. doi : 10.5802/aif.1089. https://www.numdam.org/articles/10.5802/aif.1089/

[1] J. P. Allouche et M. Mendès France, Suite de Rudin-Shapiro et modèle d'Ising, B.S.M.F., vol. 113 (1985), 273. | Numdam | MR | Zbl

[2] J. Brillhart et L. Carlitz, Note on the Shapiro polynomials, Proc. of the A.M.S., vol. 25 (1970), 114. | MR | Zbl

[3] J. Brillhart et P. Morton, On the Rudin-Shapiro polynomials, Ill. J. Math., vol. 22 (1978), 126. | Zbl

[4] G. Christol, T. Kamae, M. Mendès France et G. Rauzy, Suites algébriques, automates et substitutions, B.S.M.F., 108 (1980), 401. | Numdam | MR | Zbl

[5] I. P. Cornfeld, S. V. Fomin et Y. G. Sinai, Ergodic theory, Springer, 1982. | Zbl

[6] E. M. Coven et M. Keane, The structure of substitution minimal sets, T.A.M.S., vol. 162 (1971), 89. | MR | Zbl

[7] F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahr. Verw. Geb., vol. 41 (1978), 221. | MR | Zbl

[8] J. M. Dumont, Discrépance des progressions arithmétiques dans la suite de Morse, C.R.A.S., t. 297 (1983). | MR | Zbl

[9] P. R. Halmos, Introduction to Hilbert spaces and the theory of spectral multiplicity, Chelsea P. C. New-York, 1957. | Zbl

[10] T. Kamae, Spectral properties of automaton-generating sequences, non publié.

[11] J. Mathew et M. G. Nadkarni, A measure preserving transformation whose spectrum has Lebesgue component of multiplicity two, preprint. | Zbl

[12] J. Mathew et M. G. Nadkarni, Measure preserving transformations whose spectra have Lebesgue component of finite multiplicity, Preprint. | Zbl

[13] J.F. Mela, B. Host et F. Parreau, Analyse harmonique des mesures, Astérisque, n° 135-136 (1986). | Numdam | MR | Zbl

[14] M. Mendès France et G. Tenenbaum, Dimension des courbes planes, papiers pliés et suites de Rudin-Shapiro, B.S.M.F., vol. 109 (1981), 207. | Numdam | MR | Zbl

[15] M. Queffelec, Contribution à l'étude spectrale des suites arithmétiques, Thèse, Villetaneuse, 1984.

[16] D. Rider, Transformations of Fourier coefficients, Pacific J. Math., vol. 19 (1966), 347. | MR | Zbl

[17] W. Rudin, Some theorems on Fourier coefficients, P.A.M.S., vol. 10 (1959), 855. | MR | Zbl

[18] H. S. Shapiro, Extremal problems for polynomials and power series. M.I.T. Master's thesis, Cambridge, Mass (1951).

  • Mazáč, Jan Correlation functions of the Rudin–Shapiro sequence, Indagationes Mathematicae, Volume 35 (2024) no. 5, p. 771 | DOI:10.1016/j.indag.2023.03.003
  • Jugé, Vincent; Marcovici, Irène Finding automatic sequences with few correlations, RAIRO - Theoretical Informatics and Applications, Volume 58 (2024), p. 10 | DOI:10.1051/ita/2024008
  • Frank, Natalie Priebe; Mañibo, Neil Spectral theory of spin substitutions, Discrete and Continuous Dynamical Systems, Volume 42 (2022) no. 11, p. 5399 | DOI:10.3934/dcds.2022105
  • Jugé, Vincent; Marcovici, Irène Finding automatic sequences with few correlations, Pure Mathematics and Applications, Volume 30 (2022) no. 1, p. 75 | DOI:10.2478/puma-2022-0012
  • Tahay, Pierre-Adrien Discrete Correlation of Order 2 of Generalized Rudin-Shapiro Sequences on Alphabets of Arbitrary Size, Uniform distribution theory, Volume 15 (2020) no. 1, p. 1 | DOI:10.2478/udt-2020-0001
  • EL ABDALAOUI, E. H.; NADKARNI, M. G. A non-singular transformation whose spectrum has Lebesgue component of multiplicity one, Ergodic Theory and Dynamical Systems, Volume 36 (2016) no. 3, p. 671 | DOI:10.1017/etds.2014.85
  • BALISTER, PAUL; KALIKOW, STEVE; SARKAR, AMITES The Linus Sequence, Combinatorics, Probability and Computing, Volume 19 (2010) no. 1, p. 21 | DOI:10.1017/s0963548309990198
  • El Abdalaoui, E. H. On the spectrum of α-rigid maps, Journal of Dynamical and Control Systems, Volume 15 (2009) no. 4, p. 453 | DOI:10.1007/s10883-009-9076-x
  • Barbé, A; Haeseler, F von Correlation and spectral properties of higher-dimensional paperfolding and Rudin–Shapiro sequences, Journal of Physics A: Mathematical and General, Volume 38 (2005) no. 12, p. 2599 | DOI:10.1088/0305-4470/38/12/005
  • Allouche, Jean-Paul; Baake, Michael; Cassaigne, Julien; Damanik, David Palindrome complexity, Theoretical Computer Science, Volume 292 (2003) no. 1, p. 9 | DOI:10.1016/s0304-3975(01)00212-2
  • Queffelec, Martine Spectral study of automatic and substitutive sequences, Beyond Quasicrystals (1995), p. 369 | DOI:10.1007/978-3-662-03130-8_12
  • Wen, Zhi-Xiong; Wen, Zhi-Ying Some studies on the (p, q)-type sequences, Theoretical Computer Science, Volume 94 (1992) no. 2, p. 373 | DOI:10.1016/0304-3975(92)90045-h

Cité par 12 documents. Sources : Crossref