. Rajchman measures are those Borel measures on the circle (say) whose Fourier transform vanishes at infinity. Their study proper began with Rajchman, but attention to them can be said to have begun with Riemann's theorem on Fourier coefficients, later extended by Lebesgue. Most of the impetus for the study of Rajchman measures has been due to their importance for the question of uniqueness of trigonometric series. This motivation continues to the present day with the introduction of descriptive set theory into harmonic analysis. The last ten years have seen the resolution of several old questions, some from Rajchman himself. We give a historical survey of the relationship between Rajchman measures and their common null sets with a few of the most interesting proofs. Sommaire. Les mesures de Rajchman sont les mesures bor'eliennes sur le cercle dont la transform'ee de Fourier s'annulle `a l'infini. Leur 'etude proprement dite a commenc'e avec Rajchman, mais on peut dire qu'...

iii In memory of my father. iv

A subset of a vector space is called countably convex if it is a countable union of convex sets. Classication of countably convex subsets of topological vector spaces is addressed in this paper.

Abstract. We investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators T on separable complex F-spaces: T is frequently hypercyclic if there exists a vector x such that for every nonempty open subset U of X, thesetofintegersnsuch that T nx belongs to U has positive lower density. We give several criteria for frequent hypercyclicity, and this leads us in particular to study linear transformations from the point of view of ergodic theory. Several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting. 1.

ABSTRACT. We characterize precisely the possible rate of decay of the anti-analytic half of a trigonometric series converging to zero almost everywhere. 1.

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Abstract. In this talk, we deal with the representation of a 2π-periodic function f by a trigonometric series f(x) = a0 2 + ∞X (an cos nx + bn sin nx). n=1 A short survey will be given on this subject, with an emphasize on its history. We will touch a few aspects, namely: the origin of trigonometric series in the study of vibration and heat, the problem of the uniqueness of the trigonometric expansion and its interaction with measure theory and number theory. 1. The origin of trigonometric series We deal with the representation of a 2π-periodic function f by a trigonometric series2 f(x) = a0 2 + (an cos nx + bn sin nx). (1.1) n=1 This representation has its origin in the study of phenomena of nature: the propagation of waves and the diffusion of heat. In 1747 d’Alembert derived the wave equation ∂2u ∂t2 = v2 ∂2u ∂x2 that describes the motion of a vibrating string, where u(x, t) is the deflection of the string at position x and time t. Fourier studied heat conduction, and in 1807 gave the heat equation ∂u ∂t = k ∂2u ∂x2 which governs the diffusion of heat in a thin rod, with u(x, t) being the temperature at position x and time t. The constants v and k are physical characteristics of the string and the rod. Let us describe Fourier’s solution of the heat equation in the periodic case u(x + 2π, t) = u(x, t), which corresponds to the motion of heat in an annulus. We first look for special solutions of the form u(x, t) = ϕ(x) ψ(t). The heat equation then reads ϕ(x) ψ ′ (t) = k ϕ ′ ′ (x) ψ(t), and therefore (assuming ϕ and ψ are not identically zero) separates into ϕ ′ ′ (x) = λ ϕ(x) (1.2) and ψ ′ (t) = λk ψ(t), (1.3) where λ is a constant. Then (1.2) together with the periodicity condition ϕ(x + 2π) = ϕ(x) leads to the boundary value problem ϕ ′ ′ (x) = λ ϕ(x), ϕ(−π) = ϕ(π), ϕ ′ (−π) = ϕ ′ (π). In modern terminology, this is the eigenvalue problem for the laplace operator (the second derivative operator) on the circle T = R/2πZ. The eigenvalues are −n 2 (n = 0, 1, 2,...) and the corresponding family of eigenfunctions is the trigonometric system {1, cos nx, sin nx} (each eigenvalue has multiplicity two, except for zero 1 This is an expository talk delivered on April 10, 2007 in the Student Seminar, Tel-Aviv university (thanks to Sasha Sodin for suggesting the topic of this talk). 2 It is convenient to write the free term as a0/2 in view of formulae (2.1).

The overspill phenomenon in descriptive set theory corresponds to a forcing preservation property, with a fusion type infinite game associated to it. As an application, it is consistent with the axioms of set theory that the circle T can be covered by ℵ1 many closed sets of uniqueness while a much larger number of H-sets is necessary to cover it.

It is consistent with the axioms of set theory that the circle T can be covered by ℵ1 many closed sets of uniqueness while a much larger number of H-sets is necessary to cover it. In the proof of this theorem, the descriptive set theoretic phenomenon of overspill appears, and it is reformulated as a natural forcing preservation principle that persists through the operation of countable support product.

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