Sidon sets and Riesz products

Jean Bourgain

Displaying similar documents to “Sidon sets and Riesz products”

Fine and quasi connectedness in nonlinear potential theory

David R. Adams, John L. Lewis (1985)

Annales de l'institut Fourier

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If $B_{α, p}$ denotes the Bessel capacity of subsets of Euclidean $n$ -space, $α > 0$ , $1 < p < \infty$ , naturally associated with the space of Bessel potentials of $L^{p}$ -functions, then our principal result is the estimate: for $1 < α p \leq n$ , there is a constant $C = C (α, p, n)$ such that for any set $E$ $min {B_{α, p} (E \cap Q), B_{α, p} (E^{c} \cap Q)} \leq C \cdot B_{α, p} (Q \cap \partial_{f} E)$ for all open cubes $Q$ in $n$ -space. Here $\partial_{f} E$ is the boundary of the $E$ in the $(α, p)$ -fine topology i.e. the smallest topology on $c$ -space that makes the associated $(α, p)$ -linear potentials continuous there. As a consequence,...

$H^{p}$ Sobolev spaces

Robert S. Strichartz (1990)

Colloquium Mathematicae

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Generalized Riesz products produced from orthonormal transforms

Nikolaos Atreas, Antonis Bisbas (2012)

Colloquium Mathematicae

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Let $ℳ_{p} = {m_{k}}_{k = 0}^{p - 1}$ be a finite set of step functions or real valued trigonometric polynomials on = [0,1) satisfying a certain orthonormality condition. We study multiscale generalized Riesz product measures μ defined as weak-* limits of elements $μ_{N} \in V_{N} (N \in ℕ)$ , where $V_{N}$ are $p^{N}$ -dimensional subspaces of L₂() spanned by an orthonormal set which is produced from dilations and multiplications of elements of $ℳ_{p}$ and $\bar{⋃_{N \in ℕ} V_{N}} = L ₂ ()$ . The results involve mutual absolute continuity or singularity of such Riesz products extending previous...

Remark on the inequality of F. Riesz

W. Łenski (2005)

Banach Center Publications

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We prove F. Riesz’ inequality assuming the boundedness of the norm of the first arithmetic mean of the functions ${| φ ₙ |}^{p}$ with p ≥ 2 instead of boundedness of the functions φₙ of an orthonormal system.

Quasi-constricted linear operators on Banach spaces

Eduard Yu. Emel&#039;yanov, Manfred P. H. Wolff (2001)

Studia Mathematica

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Let X be a Banach space over ℂ. The bounded linear operator T on X is called quasi-constricted if the subspace $X ₀ : = x \in X : l i m_{n \to \infty} | | T ⁿ x | | = 0$ is closed and has finite codimension. We show that a power bounded linear operator T ∈ L(X) is quasi-constricted iff it has an attractor A with Hausdorff measure of noncompactness $χ_{| | \cdot | | ₁} (A) < 1$ for some equivalent norm ||·||₁ on X. Moreover, we characterize the essential spectral radius of an arbitrary bounded operator T by quasi-constrictedness of scalar multiples of T. Finally, we prove...

Positive bases in ordered subspaces with the Riesz decomposition property

Vasilios Katsikis, Ioannis A. Polyrakis (2006)

Studia Mathematica

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In this article we suppose that E is an ordered Banach space whose positive cone is defined by a countable family $ℱ = f_{i} | i \in ℕ$ of positive continuous linear functionals on E, i.e. E₊ = x ∈ E | $f_{i} (x) \geq 0$ for each i, and we study the existence of positive (Schauder) bases in ordered subspaces X of E with the Riesz decomposition property. We consider the elements x of E as sequences $x = (f_{i} (x))$ and we develop a process of successive decompositions of a quasi-interior point of X₊ which at each step gives elements with...

On the Fejér-F. Riesz inequality in $L^{p}$

Yoram Sagher (1977)

Studia Mathematica

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Dichotomy of global density of Riesz capacity

Hiroaki Aikawa (2016)

Studia Mathematica

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Let $C_{α}$ be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density $̲ (C_{α}, E, r) = i n f_{x \in ℝ ⁿ} C_{α} (E \cap B (x, r)) / C_{α} (B (x, r))$ for a Borel set E ⊂ ℝⁿ, where B(x,r) stands for the open ball with center at x and radius r. In case 0 < α ≤ 2, we show that $l i m_{r \to \infty} ̲ (C_{α}, E, r)$ is either 0 or 1; the first case occurs if and only if $̲ (C_{α}, E, r)$ is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also...

Quasi-algebraic representability of sets in $R^{n}$ .

W. Waliszewski (1984)

Banach Center Publications

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Construction techniques for some thin sets in duals of compact abelian groups

D. J. Hajela (1986)

Annales de l'institut Fourier

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Various techniques are presented for constructing $Λ$ (p) sets which are not $Λ (p + ϵ)$ for all $ϵ > 0$ . The main result is that there is a $Λ$ (4) set in the dual of any compact abelian group which is not $Λ (4 + ϵ)$ for all $ϵ > 0$ . Along the way to proving this, new constructions are given in dual groups in which constructions were already known of $Λ$ (p) not $Λ (p + ϵ)$ sets, for certain values of $p$ . The main new constructions in specific dual groups are: – there is a $Λ$ (2k) set which is not $Λ (2 k + ϵ)$ in $Z (2) \oplus Z (2) \oplus \dots$ for all $2 \leq k$ , $k \in N$ and...

A strong convergence theorem for H¹(𝕋ⁿ)

Feng Dai (2006)

Studia Mathematica

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Let ⁿ denote the usual n-torus and let $S ̃_{u}^{δ} (f)$ , u > 0, denote the Bochner-Riesz means of order δ > 0 of the Fourier expansion of f ∈ L¹(ⁿ). The main result of this paper states that for f ∈ H¹(ⁿ) and the critical index α: = (n-1)/2, $l i m_{R \to \infty} 1 / l o g R \int_{0}^{R} (| | S ̃_{u}^{α} (f) - {f | |}_{H ¹ (ⁿ)}) / (u + 1) d u = 0$ .

A short proof on lifting of projection properties in Riesz spaces

Marek Wójtowicz (1999)

Commentationes Mathematicae Universitatis Carolinae

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Let $L$ be an Archimedean Riesz space with a weak order unit $u$ . A sufficient condition under which Dedekind [ $σ$ -]completeness of the principal ideal $A_{u}$ can be lifted to $L$ is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of $C (X)$ -spaces. Similar results are obtained for the Riesz spaces $B_{n} (T)$ , $n = 1, 2, \dots$ , of all functions of the $n$ th Baire class on a metric space $T$ .

Some remarks on quasi-Cohen sets

Pascal Lefèvre, Daniel Li (2001)

Colloquium Mathematicae

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We are interested in Banach space geometry characterizations of quasi-Cohen sets. For example, it turns out that they are exactly the subsets E of the dual of an abelian compact group G such that the canonical injection $C (G) / C_{E^{c}} (G) ↪ L ²_{E} (G)$ is a 2-summing operator. This easily yields an extension of a result due to S. Kwapień and A. Pełczyński. We also investigate some properties of translation invariant quotients of L¹ which are isomorphic to subspaces of L¹.