Dirichlet Series and Holomorphic Functions in High Dimensions

Variants of a theorem of Helson on general Dirichlet series

“…H ∞ (λ), respectively. On the other hand, in the case of the two most prominent examples λ = (n) and λ = (log n) we have 'coincidence': D ∞ ((n)) = H ∞ ((n)) and D ∞ ((log n)) = H ∞ ((log n)); the first result is straight forward, the second one a fundamental result from [13] (see also [4,Corollary 5.3]). More generally, [5,Theorem 4.12] shows that we have the isometric 'coincidence' D ∞ (λ) = H ∞ (λ) holds, whenever • L(λ) < ∞ and D ext ∞ (λ) = D ∞ (λ) (so if e.g.…”

“…In particular, the frequency λ = (log n) satisfies Bohr's theorem which constitutes one of the fundamental tools within the theory of ordinary Dirichlet series a n n −s (see e.g. [4,Theorem 1.13,p. 21] or [20,Theorem 6.2.2.,p.…”

“…Hence, the sequence (D j k ε ) forms a Cauchy sequence in H p (λ) with limit D ε , and the proof is complete. (5) and (38) this result transfers to ordinary Dirichlet series: A Dirichlet series D = a n n −s belongs to D ∞ ((log n)) if and only if for every N its so-called Nth abschnitt, that is D| N = a n n −s , where the sum is taken over all natural numbers which only have the first N prime numbers as divisors, belong to D ∞ ((log n)) with sup N D| N ∞ < ∞ (see also [4,Corollary 3.10]).…”

“…Let H ∞ (B c 0 ) denote the Banach space of of all holomorphic and bounded functions on the open unit ball B c 0 (of the Banach space c 0 of all zero sequences). Then as proven in[13] (see also[4, Theorem 5.1]) there is an isometric bijection (38) H ∞ (B c 0 ) → H ∞ (T ∞ ), F → f,which preserves the Taylor and Fourier coefficients in the sense that c α (F ) = f (α) for all multi indices α. Recall, that F : B c 0 → C belongs to H ∞ (B c 0 ) if and only if F is continuous and all its restrictions F N : D N → C belong to H ∞ (D N ) with sup N F N ∞ < ∞ (see e.g.…”

“…In particular, f ∈ H (n) p (T) if and only if f ∈ L p (T) andf (n) = 0 for any n ∈ Z with n < 0, and f ∈ H (log n) p (T ∞ ) if and only if f ∈ L p (T ∞ ) andf (α) = 0 for any finite sequence α = (α k ) of integers with α k < 0 for some k (where as usual f (α) := f (h log p α )). Consequently, if we turn to Dirichlet series, them the Banach spaces H p = H p ((log n)) are precisely Bayart's Hardy spaces of ordinary Dirichlet series from [1] (see also [4] and [20]). 1.5.…”