Construction of functions with prescribed Hölder and chirp exponents.

Stéphane Jaffard

Revista Matemática Iberoamericana (2000)

  • Volume: 16, Issue: 2, page 331-349
  • ISSN: 0213-2230

Abstract

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We show that the Hölder exponent and the chirp exponent of a function can be prescribed simultaneously on a set of full measure, if they are both lower limits of continuous functions. We also show that this result is optimal: In general, Hölder and chirp exponents cannot be prescribed outside a set of Hausdorff dimension less than one. The direct part of the proof consists in an explicit construction of a function determined by its orthonormal wavelet coefficients; the optimality is the direct consequence of a general method we introduce in order to obtain lower bounds on the dimension of some fractal sets.

How to cite

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Jaffard, Stéphane. "Construction of functions with prescribed Hölder and chirp exponents.." Revista Matemática Iberoamericana 16.2 (2000): 331-349. <http://eudml.org/doc/39605>.

@article{Jaffard2000,
abstract = {We show that the Hölder exponent and the chirp exponent of a function can be prescribed simultaneously on a set of full measure, if they are both lower limits of continuous functions. We also show that this result is optimal: In general, Hölder and chirp exponents cannot be prescribed outside a set of Hausdorff dimension less than one. The direct part of the proof consists in an explicit construction of a function determined by its orthonormal wavelet coefficients; the optimality is the direct consequence of a general method we introduce in order to obtain lower bounds on the dimension of some fractal sets.},
author = {Jaffard, Stéphane},
journal = {Revista Matemática Iberoamericana},
keywords = {Funciones continuas; Análisis armónico; Ondículas; Fractales; Hölder exponent; chirp exponent; Hausdorff dimension; orthonormal wavelet coefficients; fractal sets},
language = {eng},
number = {2},
pages = {331-349},
title = {Construction of functions with prescribed Hölder and chirp exponents.},
url = {http://eudml.org/doc/39605},
volume = {16},
year = {2000},
}

TY - JOUR
AU - Jaffard, Stéphane
TI - Construction of functions with prescribed Hölder and chirp exponents.
JO - Revista Matemática Iberoamericana
PY - 2000
VL - 16
IS - 2
SP - 331
EP - 349
AB - We show that the Hölder exponent and the chirp exponent of a function can be prescribed simultaneously on a set of full measure, if they are both lower limits of continuous functions. We also show that this result is optimal: In general, Hölder and chirp exponents cannot be prescribed outside a set of Hausdorff dimension less than one. The direct part of the proof consists in an explicit construction of a function determined by its orthonormal wavelet coefficients; the optimality is the direct consequence of a general method we introduce in order to obtain lower bounds on the dimension of some fractal sets.
LA - eng
KW - Funciones continuas; Análisis armónico; Ondículas; Fractales; Hölder exponent; chirp exponent; Hausdorff dimension; orthonormal wavelet coefficients; fractal sets
UR - http://eudml.org/doc/39605
ER -

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