Séries de Fourier absolument convergentes

Distributions on Kronecker sets, strong forms of uniqueness, and closed ideals of A+.

“…The argument used by Helson in the case of Α(Γ) (see [21], p. 73) applies without modification to the Beurling algebra Α ω (Γ), and so there exists an order preserving one to one map φ : 0*(/V) -> ^. This concludes the proof of the corollary.…”

“…sets E for which there exists a sequence n p of distinct integers such that z" p -j^r^> l uniformly on E 9 are strong Λ^-sets and the proof of [22], theorem 3 shows that in this case ω^ρ) = l(pel) where I = J+(E). Φ {0}, but E is a ^^+-set by Wik's theorem (see [39] or [21], p. 52). To see this consider a Helson set E (i.e.…”

A quantitative version of the Beurling-Helson theorem

“…This result gives a solution of a problem of P. Lévy on endomorphisms of the algebra A(T): any such endomorphism is trivial, namely, it has the form f (t) → f (νt + t 0 ). Kahane conjectured that the weaker condition e inϕ A(T) = o(log |n|), |n| → ∞, implies that ϕ is a linear function ( [4], [5]). He showed in [5] that any continuous piecewise linear but not linear map ϕ : T → T satisfies e inϕ A (T) log |n|, |n| → ∞.…”

“…Kahane conjectured that the weaker condition e inϕ A(T) = o(log |n|), |n| → ∞, implies that ϕ is a linear function ( [4], [5]). He showed in [5] that any continuous piecewise linear but not linear map ϕ : T → T satisfies e inϕ A (T) log |n|, |n| → ∞. Thus, if the conjecture is true, then its estimate is sharp.…”

On Banach spaces of absolutely and strongly convergent Fourier series

“…We collect the well known facts about these classes in the following statement, see [1], [3], [4] or [11]: …”