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

Abstract

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.

Fourier series in $$\text {BMO}$$ with number theoretical implications

“…The following result gives a full characterization of the convergence points and shows that the divergence also occurs in irrational values extremely well approximated by rationals. Note that this sharpens [25,Th.1.4].…”

“…which is a critical case of some Fourier series considered by several authors (e.g. [16], [24], [25], [4], [3]) and, following Weierstrass, related to Riemann's strategy in the search of continuous nowhere differentiable functions. Roughly speaking we are interested in how far is this BMO function from being bounded.…”

Fourier series in BMO with number theoretical implications

“…which is a critical case of some Fourier series considered by several authors (e.g. [16], [24], [4], [3]) and, following Weierstrass, related to Riemann's strategy in the search of continuous nowhere differentiable functions.…”

Multifractal analysis of the Brjuno function

“…In contradistinction with the Hölder case, few p-spectrums have been determined: Let us mention the characteristic functions of some fractal sets [21] and random wavelet series [2]; generic results (in the Baire and prevalence settings) for functions in a Sobolev space were obtained by A. Fraysse [16]; recently, 2-exponents of trigonometric series which are not locally bounded were obtained by S. Seuret and A. Ubis [33].…”

“…Several clues indicate that the tools supplied by multifractal analysis are relevant for the Brjuno function: First it is a cocycle under the action of P GL(2, Z), as a consequence of the remarkable functional equations ∀x ∈ R \ Q, B(x + 1) = B(x), ∀x ∈ (0, 1) \ Q, B(x) = log(1/x) + xB(1/x), see [24,25]. This property is reminiscent of the behavior of the Jacobi theta function under modular transforms, which is the key ingredient in the determination of the pointwise exponent of the non-differentiable Riemann function R(x) = sin(πn 2 x)/n 2 [18], and of related trigonometric series [33]. Other trigonometric series also related to modular forms have been studied by I. Petrykiewicz in [30,31].…”