Translations mesurables et ensembles de Rosenthal
Trigonometric sums associated with pseudo-measures
Two examples of subspaces in spanned by characters of finite order
By a Fourier multiplier technique on Cantor-like Abelian groups with characters of finite order, the norms from L² into of certain embeddings of character sums are computed. It turns out that the orders of the characters are immaterial as soon as they are at least four.
Two random constructions inside lacunary sets
We study the relationship between the growth rate of an integer sequence and harmonic and functional properties of the corresponding sequence of characters. In particular we show that every polynomial sequence contains a set that is for all but is not a Rosenthal set. This holds also for the sequence of primes.
UC-sets in the dual object of compact groups
Une classe d'ensembles d'unicité des séries trigonométriques
Uniqueness of complete norms for quotients of Banach function algebras
We prove that every quotient algebra of a unital Banach function algebra A has a unique complete norm if A is a Ditkin algebra. The theorem applies, for example, to the algebra A (Γ) of Fourier transforms of the group algebra of a locally compact abelian group (with identity adjoined if Γ is not compact). In such algebras non-semisimple quotients arise from closed subsets E of Γ which are sets of non-synthesis. Examples are given to show that the condition of Ditkin cannot be relaxed. We construct...
ε-Kronecker and I₀ sets in abelian groups, III: interpolation by measures on small sets
Let U be an open subset of a locally compact abelian group G and let E be a subset of its dual group Γ. We say E is I₀(U) if every bounded sequence indexed by E can be interpolated by the Fourier transform of a discrete measure supported on U. We show that if E·Δ is I₀ for all finite subsets Δ of a torsion-free Γ, then for each open U ⊂ G there exists a finite set F ⊂ E such that E∖F is I₀(U). When G is connected, F can be taken to be empty. We obtain a much stronger form of that for Hadamard sets...
ε-Kronecker and I₀ sets in abelian groups, IV: interpolation by non-negative measures
A subset E of a discrete abelian group is a "Fatou-Zygmund interpolation set" (FZI₀ set) if every bounded Hermitian function on E is the restriction of the Fourier-Stieltjes transform of a discrete, non-negative measure. We show that every infinite subset of a discrete abelian group contains an FZI₀ set of the same cardinality (if the group is torsion free, a stronger interpolation property holds) and that ε-Kronecker sets are FZI₀ (with that stronger interpolation property). ...
О сходимости почти всюду рядов Фурье по ортонормированным системам типа Франклина