If F is a C3+γ area-preserving surface diffeomorphism with an invariant curve that is a C2+β topological circle Chttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332838/8bb86808-2271-4123-9a6c-b7bf13a7da85/content/eq2165.tif"/> on which F acts like a rotation by a "typical" irrational rotation number, then Chttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332838/8bb86808-2271-4123-9a6c-b7bf13a7da85/content/eq2166.tif"/> is in fact C2+γ′ for all γ′ < γ.

Given a continuous (or just measurable) real-valued function on [0, 1] and a closed subset E ⊂ [0, 1], denote by f | E the restriction of f to E . f | E can be “better behaved” than f and we discuss the existence, for every f , or every f in some class, of sets E that are substantial in terms of fractal dimension, such that f | E has bounded total variation, or is monotone, or satisfies a given modulus of continuity.

Cet article a pour but de montrer le lien entre la méthode introduite en 1959 par Paul Malliavin pour démontrer l'impossibilité de la synthèse spectrale sur les groupes abéliens non compacts [Paul Malliavin, Sur l'impossibilité de la synthèse spectrale sur la droite, C. R. Acad. Sci. Paris 248 (1959) 2155–2157 ; Paul Malliavin, Sur l'impossibilité de la synthèse spectrale sur les groupes abéliens non compacts, Publ. Math. Inst. Hautes Etudes Sci. 2 (1959) 61–68] et une recherche actuelle sur les propriétés des restrictions des fonctions continues [Jean-Pierre Kahane, Yitzhak Katznelson, Restrictions of continuous functions, Israel J. Math., à paraître]. Il apportera un complément sur ce dernier sujet.

This is the material for two lectures given at Ecole Polytechnique in May 2011 for the math teachers of "classes préparatoires"(parallel to the undergraduate classes in universities). The introduction is a personal overview on Fourier analysis, its history, the terms in use and a few references. The first part is also an overview but on a restricted subject : the link between statistics and Fourier, from Legendre and the role of l^2 to Donoho and the role of l^1. The third part starts from a theorem of Candès, Romberg and Tao on compressive sampling and applies the basic idea to a quite different question : the reconstruction of a function whose unknown spectrum has large gaps from Its restriction to an interval. The description and extention of the method of Candès, Romberg and Tao in Appendix 2 was developed later in a note aux Comptes rendus and an article submitted to Annales de l'Institut Fourier.

Random sequences of integers, Sidon sets, density in the Bohr group, and sets of analyticity. We study properties of a sequence A obtained by a random selection of integers n, where n epsilon Lambda with probability pi(n) independently of the other choices. We distinguish two cases: if lim sup(n ->infinity)n pi(n) < infinity, A is a.s. a Sidon set, non-dense in the Bohr group; if lim(n ->infinity) n pi(n) = infinity, then Lambda is a.s. a set of analyticity and is dense in the Bohr group.

Vector spaces Linear operators and matrices Duality of vector spaces Determinants Invariant subspaces Operators on inner-product spaces Structure theorems Additional topics Appendix Index Symbols.

1 WHEN IS A SEQUENCE DENSE IN A COMPACT ABELIAN GROUP We denote by G a compact abelian group; we denote by μ its normalized Haar measure and by Γ = Ĝ its dual group (which is discrete). We consider sequences Λ ⊂ G and ask when is Λ dense in G. Our first approach is based on the duality theorem which tells us that the topology on G is the weak topology determined by Γ; thus if g0 ∈ G, then a basis for the neighborhoods of g0 in G is given by the sets ⋂k j=1{g : |〈g,γ j〉−〈g0,γ j〉| 0. This proves Lemma 1.1. Λ is dense in G if, and only if, given k ∈ Z+, γ1 · · ·γk ∈ Γ, g0 ∈G, and e > 0, there exists λ ∈ Λ such that

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Introduction Let 0 < β < 1 and let φi(t) = βt+αi, i = 1, 2, . . . , r, be r affine contractions of the unit interval [0, 1] into itself. We assume 0 = α1 < α2 < α3 < · · · < αr = 1 − β. If in addition αi+1 − αi ≤ β for i = 1, 2, . . . , r − 1 so that φi+1(0) ≤ φi(1), then the union of the images of [0, 1] covers [0, 1], and for every t ∈ [0, 1] there exists at least one pair (i′, t′) such that t = φi′(t). Setting for any t ∈ [0, 1],