Une nouvelle propriété des suites de Rudin-Shapiro

Martine Queffelec

Annales de l'institut Fourier (1987)

  • Volume: 37, Issue: 2, page 115-138
  • ISSN: 0373-0956

Abstract

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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 q φ ( q ) .

How to cite

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Queffelec, Martine. "Une nouvelle propriété des suites de Rudin-Shapiro." Annales de l'institut Fourier 37.2 (1987): 115-138. <http://eudml.org/doc/74748>.

@article{Queffelec1987,
abstract = {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 $q\varphi (q)$.},
author = {Queffelec, Martine},
journal = {Annales de l'institut Fourier},
keywords = {Rudin-Shapiro sequences; extremal properties; automaton sequence; spectrum; maximal spectral type; spectral multiplicity; multiplicity function; q-generalized Rudin-Shapiro sequence; substitution; dynamical system; Riesz products},
language = {fre},
number = {2},
pages = {115-138},
publisher = {Association des Annales de l'Institut Fourier},
title = {Une nouvelle propriété des suites de Rudin-Shapiro},
url = {http://eudml.org/doc/74748},
volume = {37},
year = {1987},
}

TY - JOUR
AU - Queffelec, Martine
TI - Une nouvelle propriété des suites de Rudin-Shapiro
JO - Annales de l'institut Fourier
PY - 1987
PB - Association des Annales de l'Institut Fourier
VL - 37
IS - 2
SP - 115
EP - 138
AB - 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 $q\varphi (q)$.
LA - fre
KW - Rudin-Shapiro sequences; extremal properties; automaton sequence; spectrum; maximal spectral type; spectral multiplicity; multiplicity function; q-generalized Rudin-Shapiro sequence; substitution; dynamical system; Riesz products
UR - http://eudml.org/doc/74748
ER -

References

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  1. [1] J. P. ALLOUCHE et M. MENDÈS FRANCE, Suite de Rudin-Shapiro et modèle d'Ising, B.S.M.F., vol. 113 (1985), 273. Zbl0592.10049MR87h:11020
  2. [2] J. BRILLHART et L. CARLITZ, Note on the Shapiro polynomials, Proc. of the A.M.S., vol. 25 (1970), 114. Zbl0191.35101MR41 #5575
  3. [3] J. BRILLHART et P. MORTON, On the Rudin-Shapiro polynomials, Ill. J. Math., vol. 22 (1978), 126. Zbl0371.10009
  4. [4] G. CHRISTOL, T. KAMAE, M. MENDÈS FRANCE et G. RAUZY, Suites algébriques, automates et substitutions, B.S.M.F., 108 (1980), 401. Zbl0472.10035MR82e:10092
  5. [5] I. P. CORNFELD, S. V. FOMIN et Y. G. SINAI, Ergodic theory, Springer, 1982. Zbl0493.28007
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  7. [7] F. M. DEKKING, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahr. Verw. Geb., vol. 41 (1978), 221. Zbl0348.54034MR57 #1455
  8. [8] J. M. DUMONT, Discrépance des progressions arithmétiques dans la suite de Morse, C.R.A.S., t. 297 (1983). Zbl0533.10005MR85f:11058
  9. [9] P. R. HALMOS, Introduction to Hilbert spaces and the theory of spectral multiplicity, Chelsea P. C. New-York, 1957. Zbl0079.12404
  10. [10] T. KAMAE, Spectral properties of automaton-generating sequences, non publié. Zbl0371.10044
  11. [11] J. MATHEW et M. G. NADKARNI, A measure preserving transformation whose spectrum has Lebesgue component of multiplicity two, preprint. Zbl0515.28010
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  13. [13] J.F. MELA, B. HOST et F. PARREAU, Analyse harmonique des mesures, Astérisque, n° 135-136 (1986). Zbl0589.43001MR88a:43005
  14. [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. Zbl0468.10033MR82k:10073
  15. [15] M. QUEFFELEC, Contribution à l'étude spectrale des suites arithmétiques, Thèse, Villetaneuse, 1984. 
  16. [16] D. RIDER, Transformations of Fourier coefficients, Pacific J. Math., vol. 19 (1966), 347. Zbl0144.32001MR34 #3195
  17. [17] W. RUDIN, Some theorems on Fourier coefficients, P.A.M.S., vol. 10 (1959), 855. Zbl0091.05706MR22 #6979
  18. [18] H. S. SHAPIRO, Extremal problems for polynomials and power series. M.I.T. Master's thesis, Cambridge, Mass (1951). 

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