$$\pmb {{\mathcal {H}}_{p}}$$-Theory of General Dirichlet Series

Vector-valued general Dirichlet series

“…In [5] Bayart developed an H p -theory of ordinary Dirichlet series for 1 ≤ p ≤ ∞. This was later extended to λ-Dirichlet series in [28] through Fourier analysis on groups. Providing a vector-valued definition is then straightforward and gives rise to the spaces H p (λ, X) and H + p (λ, X), which are properly defined in Section 2.…”

“…The spaces H p (λ, X). From [28] (see also [31]) we recall the definition of and some basic facts about Dirichlet groups and we refer to [57] for background on Fourier analysis on groups. Let G be a compact abelian group and β : (R, +) → G a homomorphism of groups.…”

“…From [28,Section 3.1] we know that every L 1 (R)-function may be interpreted as a bounded regular Borel measure on G. In particular, for every u > 0 the Poisson kernel P u (t) := 1 π u u 2 + t 2 , t ∈ R, defines a measure p u on G, which we call the Poisson measure on G. We have p u ≤ P u L 1 (R) = 1 and p u (h x ) = P u (x) = e −u|x| for all u > 0 and x ∈ β( G). Now, given a frequency λ, we call a Dirichlet group (G, β) a λ-Dirichlet group whenever λ ⊂ β( G), or equivalently for every e −iλn• ∈ (R, +) there is (a unique) h λn ∈ G with h λn • β = e −iλn• .…”

“…exists, since we integrate an almost periodic function on R, and it defines a norm on the space Pol(λ, X) of all X-valued λ-Dirichlet polynomials. Then H p (λ, X) is the completion of (Pol(λ, X), • p ); here the density of Pol(λ, X) in H p (λ, X) follows by an analysis of the arguments in the scalar case, given in [28,Proposition 3.14]. For completeness we sketch the proof: Fix some Dirichlet group (G, β) and A ⊂ G. Denote by C A (G, X) the Banach space of all continuous functions f : G → X with Fourier transform supported on A, and by Pol A (G, X) the set of all X-valued polynomials of the form γ∈A x γ γ.…”

Henry Helson meets other big shots — a brief survey

“…Finally, we recall that every function f ∈ L 1 (T ∞ ) for almost all z ∈ T ∞ allows a locally Lebesgue integrable 'restriction' f z : R → C such that f z (t) = f (zβ(t)) for almost all t ∈ R (see [10,Lemma 3.10]). More explicitly, for almost all z ∈ T ∞ the function…”