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...

Le théorème classique de Riesz-Raikov assure que, pour tout entier $\theta \&gt;1$ et toute $f$ de $L^1\left(𝕋\right)$, où $𝕋=ℝ/\mathbb{Z}$, les moyennes$$\frac{1}{N}\sum _1^{N}f\left({\theta }^{n}x\right)\text{convergent}\text{vers}{\int }_{𝕋}f\left(t\right)\mathrm{d}t$$pour presque tout point $x$ de $ℝ$. J.Bourgain (cf.Israël Math. Conf. Proc. 1990) a prouvé que la convergence précédente a lieu pour tout réel algébrique $\theta \&gt;1$ et toute $f$ de $L^2\left(𝕋\right)$. Dans cet article nous prouvons que, si $\varphi$ est un endomorphisme de ${ℝ}^p$ algébrique sur $\mathbb{Q}$, dont les valeurs propres sont toutes de module $\&gt;1$, alors pour toute $f$ de $L^2\left({𝕋}^p\right)$, les moyennes $(1/N){\sum }_1^{N}f\left({\varphi }^{n}x\right)$ convergent vers ${\int }_{{𝕋}^p}f\left(t\right)\mathrm{d}t$ pour presque tout point $x$ de ${ℝ}^p$. Nous...

We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class $Lip(\alpha ₁,...,{\alpha }_N)$ for some $0<\alpha ₁,...,{\alpha }_N<1$ and its marginal functions satisfy $f(·,x₂,...,x_N)\in Lip\beta ₁,...,f(x₁,...,x_{N-1},·)\in Lip{\beta }_N$ for some $0<\beta ₁,...,{\beta }_N<1$ uniformly in the indicated variables $x_l$, 1 ≤ l ≤ N, then $f{̃}^{(\eta ₁,...,{\eta }_N)}\in Lip(\alpha ₁,...,{\alpha }_N)$ for each choice of $(\eta ₁,...,{\eta }_N)$ with ${\eta }_l=0$ or 1 for 1 ≤ l ≤ N.

We extend the results of paper of F. Móricz (2010), where necessary conditions were given for the $L^1$-convergence of double Fourier series. We also give necessary and sufficient conditions for the $L^1$-convergence under appropriate assumptions.

We give necessary conditions in terms of the coefficients for the convergence of a double trigonometric series in the $L^p$-metric, where $0<p<1$. The results and their proofs have been motivated by the recent papers of A. S. Belov (2008) and F. Móricz (2010). Our basic tools in the proofs are the Hardy-Littlewood inequality for functions in $H^p$ and the Bernstein-Zygmund inequalities for the derivatives of trigonometric polynomials and their conjugates in the $L^p$-metric, where $0<p<1$.

Multi-dimensional generalizations of the Wiener-Żelazko and Lévy-Żelazko theorems are obtained.

A corona type theorem is given for the ring D'A(Rd) of periodic distributions in Rd in terms of the sequence of Fourier coefficients of these distributions,which have at most polynomial growth. It is also shown that the Bass stable rank and the topological stable rank of D'A(Rd) are both equal to 1.