Multifractal behavior of polynomial Fourier series

Chamizo, Fernando; Ubis, Adrián

doi:10.1016/j.aim.2013.09.015

Cited by 18 publications

(21 citation statements)

References 34 publications

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“…In the second sum, there is at most one integer m for which |N m/q − 2N 2 h| < N/2q, and the corresponding term is bounded above by 3 and u k = 2 −2k N/q. The sum over k of this upper bound is finite, and this sum can be bounded above independently on N and q.…”

Section: 5mentioning

confidence: 99%

Local L^2-regularity of Riemann’s Fourier series

Seuret¹,

Ubis²

2017

Ann. inst. Fourier

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We are interested in the convergence and the local regularity of the lacunary Fourier series Fs(x) = +∞ n=1 e 2iπn 2 x n s. In the 1850's, Riemann introduced the series F2 as a possible example of nowhere differentiable function, and the study of this function has drawn the interest of many mathematicians since then. We focus on the case when 1/2 < s ≤ 1, and we prove that Fs(x) converges when x satisfies a Diophantine condition. We also study the L 2local regularity of Fs, proving that the local L 2 -norms have different behaviors around different x, according again to Diophantine conditions on x.

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Section: 5mentioning

confidence: 99%

Local L^2-regularity of Riemann’s Fourier series

Seuret¹,

Ubis²

2017

Ann. inst. Fourier

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“…Le spectre obtenu est donné figure 5. L'analyse multifractale des séries de Fourier est toujours active (voir [15,27], ou [9] pour un point de vue un peu différent), par exemple on ne connait Pour les mesureségalement on peut définir un formalisme multifractal.…”

Section: 2unclassified

Quelques résultats d’analyse multifractale en analyse

Seuret¹

2014

Séminaire Laurent Schwartz — EDP Et Applications

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“…Then, by (18) with α = 0 d , (13). Further, recalling that the function f ∈ C η (R d ), the wavelet coefficients of f j satisfy |d λ | 2 −jη , one sees that f j ∞ 2 −jη (here the assumption that the wavelets are compactly supported makes the computations easier).…”

Section: With Uniform Constants) This Yields the Upper Boundmentioning

confidence: 99%

Multifractal Analysis and Wavelets

Seuret¹

2016

New Trends in Applied Harmonic Analysis

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Abstract. In this course, we give the basics of the part of multifractal theory that intersects wavelet theory. We start by characterizing the pointwise Hölder exponents by some decay rates of wavelet coefficients. Then, we give some examples of wavelet series having a multifractal behavior, and we explain how to build wavelet series with prescribed pointwise Hölder exponents. Next we develop the problematics of multifractal formalism, going from the intuitive formula by Frisch and Parisi to explicit and exploitable formulas. We prove that "multifractals are everywhere", in the sense that typical functions in Besov spaces or typical measures are multifractal in the sense of Baire's categories. We finish by some well-known examples of multifractal wavelet series, random and deterministic, focusing on the influence of certain adaptive threshold procedures to the multifractal properties of signals.

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Multifractal behavior of polynomial Fourier series

Abstract: Esta es la versión de autor del artículo publicado en: This is an author produced version of a paper published in:

Cited by 18 publications

References 34 publications

Local L^2-regularity of Riemann’s Fourier series

Local L^2-regularity of Riemann’s Fourier series

Quelques résultats d’analyse multifractale en analyse

Multifractal Analysis and Wavelets

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