We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions ${\Delta }_{h}f\left(x\right)=f(x+h)-f\left(x\right)$ are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, $ℌ(ℱ,G)=H\subset ℝ/\mathbb{Z}:\left(\exists f\in ℱG\right)\left(\forall h\in H\right){\Delta }_{h}f\in G$, we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group $𝕋=ℝ/\mathbb{Z}$ that are invariant for changes on null-sets (e.g. measurable...

We consider sequences of linear operators Uₙ with a localization property. It is proved that for any set E of measure zero there exists a set G for which $U{ₙ}_G\left(x\right)$ diverges at each point x ∈ E. This result is a generalization of analogous theorems known for the Fourier sum operators with respect to different orthogonal systems.

Soit $𝒰$ (resp. ${𝒰}_0$) l’ensemble des compacts d’unicité (resp. d’unicité au sens large) du tore $\mathbf{T}$. On montre qu’un borélien de $\mathbf{T}$ dont tout sous-compact est dans ${𝒰}_0$ est nécessairement contenu dans une réunion dénombrable de compacts de ${𝒰}_0$, et on montre que cette propriété n’est plus vraie quand on remplace ${𝒰}_0$ par $𝒰$.Comme conséquence on obtient qu’un borélien qui est d’unicité est nécessairement maigre. On en déduit aussi l’existence d’un compact d’unicité qui ne peut être recouvert par une suite de compacts...

Upper bounds for GCD sums of the form ${\sum }_{k,\ell =1}^N\frac{{\left(\gcd (n_k,n_{\ell })\right)}^{2\alpha }}{{\left(n_k{n}_{\ell }\right)}^{\alpha }}$ are established, where ${\left(n_k\right)}_{1\le k\le N}$ is any sequence of distinct positive integers and $0<\alpha \le 1$; the estimate for $\alpha =1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $\alpha =1/2$. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish...

Let $s\in \mathbf{C}$, $x\in \mathbf{R}/2\pi \mathbf{Z}$. We construct Dirichlet series $F(x,x)$ where for each fixed $s$ in a half plane, $\mathrm{Re}F(x,x)$, as a function of $x$, is a non-synthesizable absolutely convergent Fourier series. Because of the way the frequencies in $F$ are chosen, we are motivated to introduce a class of synthesizable absolutely convergent Fourier series which are defined in terms of idele characters. We solve the “problem of analytic continuation” in this setting by constructing pseudo-measures, determined by idele characters, when $\mathrm{Re}s\le 1$.