Sidon proved the inequality named after him in 1939. It is an upper estimate for the integral norm of a linear combination of trigonometric Dirichlet kernels expressed in terms of the coefficients. Since the estimate has many applications for instance in $L^1$ convergence problems and summation methods with respect to trigonometric series, newer and newer improvements of the original inequality has been proved by several authors. Most of them are invariant with respect to the rearrangement of the coefficients....

Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with $\Sigma \subset {ℝ}^{n-1}$, a bounded domain of class $C^{1,1}$, are obtained in the space $L^q(ℝ;L²\left(\Sigma \right))$, q ∈ (1,∞). As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and R-boundedness of operator families.

For 1 ≤ q < ∞, let ${}_q\left(\right)$ denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes ${}_q\left(\right)$ as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction q >...

The main observation of this note is that the Lebesgue measure μ in the Turán-Nazarov inequality for exponential polynomials can be replaced with a certain geometric invariant ω ≥ μ, which can be effectively estimated in terms of the metric entropy of a set, and may be nonzero for discrete and even finite sets. While the frequencies (the imaginary parts of the exponents) do not enter the original Turán-Nazarov inequality, they necessarily enter the definition of ω.

We prove a class of uncertainty principles of the form $\parallel S_{g}f{\parallel }_1\le C(\parallel x^{a}f{\parallel }_p+\parallel {\omega }^{b}f̂{\parallel }_q)$, where $S_{g}f$ is the short time Fourier transform of f. We obtain a characterization of the range of parameters a,b,p,q for which such an uncertainty principle holds. Counter-examples are constructed using Gabor expansions and unimodular polynomials. These uncertainty principles relate the decay of f and f̂ to their behaviour in phase space. Two applications are given: (a) If such an inequality holds, then the Poisson summation formula is valid...

En dimension 1 on analyse la fonction irrégulière $r\left(x\right)={\sum }_{n=1}^{\infty }{n}^{-p}sin\left(n^{p}x\right)$ (p entier ≥ 2) en un point $x_0$ de dérivabilité (π est un tel point) et on démontre que le terme d’erreur est un chirp de classe (1 + 1/(2p-2), 1/(p-1), (p-1)/p). La fonction r(x) est dans l’espace 2-microlocal $C_{x_0}^{s,s^{\text{'}}}$ si et seulement si s+s’ ≤ 1 - 1/p et ps+s’≤ p - 1/2. En dimension 2, on obtient en (π,π) l’existence d’un plan tangent pour la surface $z={\sum }_{m,n=1}^{\infty }{(m^2+n^2)}^{-\gamma }sin(m^{2}x+n^{2}y)$ dès que γ>1.

Nilsequences arose in the study of the multiple ergodic averages associated to Furstenberg’s proof of Szemerédi’s Theorem and have since played a role in problems in additive combinatorics. Nilsequences are a generalization of almost periodic sequences and we study which portions of the classical theory for almost periodic sequences can be generalized for two step nilsequences. We state and prove basic properties for two step nilsequences and give a classification scheme for them.

Constitutive equations of continuum mechanics of the solid phase of anisotropic material is focused in the paper. First, a synoptic one-dimensional Maxwell model is explored, subjected to arbitrary deformation load. The explicit form is derived for stress on strain dependence. Further, the analogous explicit constitutive equation is taken in three spatial dimensions and treated mathematically. Later on, a simply supported straight concrete beam reinforced by the steel fibres is taken as an investigated...

If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of $L^p$ and weak $L^p$ boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces $L^{\Phi }$ having the property $L^{\infty }\subset L^{\Phi }\subset L^p$, $1\le p<\infty$. The second contains spaces $L^{\Phi }$ that...

We investigate the approximation properties of the partial sums of the Fourier series and prove some direct and inverse theorems for approximation by polynomials in weighted Orlicz spaces. In particular we obtain a constructive characterization of the generalized Lipschitz classes in these spaces.

Soient $G_1$ et $G_2$ deux groupes abéliens localement compacts de dual ${\Gamma }_1$ et ${\Gamma }_2$. Soit $h:{\Gamma }_1\to {\Gamma }_2$ un homomorphisme continu d’image dense de ${\Gamma }_1$ dans ${\Gamma }_2$. Soit $1\le p\le \infty$ ; on prouve un théorème d’approximation des multiplicateurs de $\mathbf{F}{L}^p(G_2)$ et on utilise ce résultat pour démontrer le suivant : soit $m:{\Gamma }_2\to \mathbf{C}$ une fonction continue ; $m$ est un multiplicateur de $\mathbf{F}{L}^p(G_2)$ si, et seulement si, $m\circ h$ est un multiplicateur de $\mathbf{F}{L}^p(G_1)$.

The direct and inverse problems of approximation theory in the subspace of weighted generalized grand Lebesgue spaces of 2π-periodic functions with the weights satisfying Muckenhoupt's condition are investigated. Appropriate direct and inverse theorems are proved. As a corollary some results on constructive characterization problems in generalized Lipschitz classes are presented.