On étudie la décroissance à l’infini des coefficients de Fourier des fonctions $2\pi$-périodiques intégrables. Soit en particulier ${\lambda }_n$ une suite lacunaire d’entiers : ${\lambda }_{n+1}\ge 3{\lambda }_n$. On appelle suite $k$-lacunaire associée la suite ${\mu }_N^k$ des entiers qui s’écrivent sous la forme $\pm {\lambda }_{n_1}\pm {\lambda }_{n_2}\pm \cdots \pm {\lambda }_{n_k}$, $n_1\>n_2\>\cdots \>n_k$. On montre que si ${\int }_0^{2\pi }\left|f\right|({\mathrm{Log}}^{+}\left|f\right|{)}^{k/2}dx$ est fini, il en est de même de ${\sum }_N|\widehat{f}({\mu }_N^k){|}^2$. D’autre part, si ${\lambda }_n$ satisfait à une condition plus restrictive, quel que soit $1\<p\le 2$, si ${\int }_0^{2\pi }|f{|}^{p}dx$ est fini il en est de même de ${\sum }_k(p-1){\sum }_N|\widehat{f}({\mu }_N^k){|}^2$. Ces résultats sont généralisés à d’autres groupes que $\mathbf{R}/2\pi \mathbf{Z}$, et à d’autres...

A set S of integers is called ε-Kronecker if every function on S of modulus one can be approximated uniformly to within ε by a character. The least such ε is called the ε-Kronecker constant, κ(S). The angular Kronecker constant is the unique real number α(S) ∈ [0,1/2] such that κ(S) = |exp(2πiα(S)) - 1|. We show that for integers m > 1 and d ≥ 1, $\alpha 1,m,...,m^{d-1}=(m^{d-1}-1)/\left(2(m^d-1)\right)$ and α1,m,m²,... = 1/(2m).

Christensen has defined a generalization of the property of being of Haar measure zero to subsets of (abelian) Polish groups which need not be locally compact; a recent paper of Hunt, Sauer, and Yorke defines the same property for Borel subsets of linear spaces, and gives a number of examples and applications. The latter authors use the term “shyness” for this property, and “prevalence” for the complementary property. In the present paper, we construct a number of examples of non-shy Borel sets...

The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations $\frac{d}{dt}[x\left(t\right)-L\left(x_t\right)]=A[x\left(t\right)-L\left(x_t\right)]+G\left(x_t\right)+\frac{1}{\Gamma \left(\alpha \right)}{\int }_{-\infty }^t{(t-s)}^{\alpha -1}\left({\int }_{-\infty }^{s}a(s-\xi )x\left(\xi \right)d\xi \right)ds+f\left(t\right)$, ($\alpha >0$) with the periodic condition $x\left(0\right)=x\left(2\pi \right)$, where $a\in L^1\left({ℝ}_{+}\right)$ . Our approach is based on the R-boundedness of linear operators $L^p$-multipliers and UMD-spaces.

Let G be a compact abelian group with dual group Γ and let ε > 0. A set E ⊂ Γ is a “weak ε-Kronecker set” if for every φ:E → there exists x in the dual of Γ such that |φ(γ)- γ(x)| ≤ ε for all γ ∈ E. When ε < √2, every bounded function on E is known to be the restriction of a Fourier-Stieltjes transform of a discrete measure. (Such sets are called I₀.) We show that for every infinite set E there exists a weak 1-Kronecker subset F, of the same cardinality as E, provided there are not “too many”...

Let p,q,n be natural numbers such that p+q = n. Let be either ℂ, the complex numbers field, or ℍ, the quaternionic division algebra. We consider the Heisenberg group N(p,q,) defined ⁿ × ℑ , with group law given by (v,ζ)(v’,ζ’) = (v + v’, ζ + ζ’- 1/2 ℑ B(v,v’)), where $B(v,w)={\sum }_{j=1}^p{v}_j\overline{w_j}-{\sum }_{j=p+1}^n{v}_j\overline{w_j}$. Let U(p,q,) be the group of n × n matrices with coefficients in that leave the form B invariant. We compute explicit fundamental solutions of some second order differential operators on N(p,q,) which are canonically associated to...

Consider a simple non-compact algebraic group, over any locally compact non-discrete field, which has Kazhdan’s property $\left(T\right)$. For any such group, $G$, we present a Kazhdan set of two elements, and compute its best Kazhdan constant. Then, settling a question raised by Serre and by de la Harpe and Valette, explicit Kazhdan constants for every lattice $\Gamma$ in $G$ are obtained, for a “geometric” generating set of the form $\Gamma \cap B_r$, where $B_r\subset G$ is a ball of radius $r$, and the dependence of $r$ on $\Gamma$ is described explicitly....

On munit la classe des algèbres de Kac d’une nouvelle classe de morphismes, stable par dualité. Cela permet de rendre compte, dans les cas abélien ou symétrique, de la catégorie des groupes localement compacts munis des morphismes continus de groupe. Le lien avec les morphismes précédemment définis et beaucoup plus restrictifs est établi.