Vector-valued general Dirichlet series

Carando, Daniel; Defant, Andreas; Marceca, Felipe; Schoolmann, Ingo

doi:10.4064/sm200127-24-4

Cited by 9 publications

(17 citation statements)

References 53 publications

(148 reference statements)

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“…More background. Mainly inspired by the classical monograph [13] of Hardy and M. Riesz from 1915, the recent works [2], [6], [8], [9], [10], [17], and [20] suggest a modern study of general Dirichlet series D = a n e −λns . Whereas the dominant tool of the early days of this theory was complex analysis, the idea now is to implement modern techniques like functional analysis, Fourier analysis, or abstract harmonic analysis on compact abelian groups.…”

Section: Introductionmentioning

confidence: 99%

“…For ordinary Dirichlet series these spaces have been introduced in [1] and [14], and this has in fact caused a fruitful renaissance of the analysis of such series. Within general Dirichlet series various aspects of H p (λ) have been studied in [2], [6], [8], [9], [10], and [20].…”

Section: Introductionmentioning

confidence: 99%

“…But then in view of Bohr's approximation theorem for uniformly almost periodic functions on R, and in view of the approximation theorem from (6) for almost periodic holomorphic functions on [Re > 0], the following question arises:…”

Section: Introductionmentioning

confidence: 99%

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Holomorphic functions of finite order generated by Dirichlet series

Defant¹,

Schoolmann²

2021

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Given a frequency λ = (λ n ) and ℓ ≥ 0, we introduce the scale of Banach spaces H λ ∞,ℓ [Re > 0] of holomorphic functions f on the right half-plane [Re > 0], which satisfy (A) the growth condition |f (s)| = O((1 + |s|) ℓ ), and (B) are such that they on some open subset and for some m ≥ 0 coincide with the pointwise limit (as x → ∞) of the so-called (λ, m)-Riesz means λn 0of some λ-Dirichlet series a n e −λns . Reformulated in our terminology, an important result of M. Riesz shows that in this case the function f for every k > ℓ is the pointwise limit of the (λ, k)-Riesz means of D.Our main contribution is an extension -showing that 'after translation' every bounded set in H λ ∞,ℓ [Re > 0] is uniformly approximable by all its (λ, k)-Riesz means of order k > ℓ. This follows from an appropriate maximal theorem, which in fact turns out to be at the very heart of a seemingly interesting structure theory of the Banach spacesℓ [Re > 0] basically consists of those holomorphic functions on [Re > 0], which fulfill the growth condition |f (s)| = O((1 + |s|) ℓ ) and are of finite uniform order on all abscissas [Re = σ], σ > 0.To establish all this and more, we need to reorganize (and to improve) various aspects and keystones of the classical theory of Riesz summability of general Dirichlet series as invented by Hardy and M. Riesz. DEFANT, SCHOOLMANN, AND ANDREAS DEFANT AND INGO SCHOOLMANN 4.1. Ordinary and power case 29 4.2. Perron formula -a variant 30 4.3. A maximal inequality for Riesz means 31 4.4. Uniform Riesz approximation 32 4.5. Theorem 41 of M. Riesz revisted 33 4.6. The bounded case ℓ = 0 34 4.7. Boundedness of coefficient functionals 34 4.8. A Montel theorem 35 4.9. Completeness 36 4.10. Behavior far left 37 4.11. Behavior far right 37 4.12. Finite order 39 4.13. Equivalence 40 References 44

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Holomorphic functions of finite order generated by Dirichlet series

Defant¹,

Schoolmann²

2021

Preprint

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“…The results on ordinary Dirichlet series which we just indicated, will be performed within a setting of general Dirichlet series -a far more challengeing task. Fixing a frequency λ, an extensive 'modern' study of λ-Dirichlet series a n e −λns has been started in the recent articles [2], [8], [9], [10], [11], [12], [24], [25], and one of the major concepts is the introduction of so-called Hardy spaces H p (λ) of λ-Dirichlet series (in Section 6 we repeat the definition).…”

mentioning

confidence: 99%

“…As indicated in the preceding section the frequency λ = (log n) satisfies Bohr's theorem. A delicate question, which came up in [12], then is whether we have a n e −itλn (for the third equivalence see [8,Theorem 4.12]). Counterexamples for (13) are then provided by [25,Theorem 5.2], and concrete sufficient conditions on the frequency λ were given by Bohr [6], Landau [22], and more recently Bayart [2].…”

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confidence: 99%

Riesz summability on boundary lines of holomorphic functions generated by Dirichlet series

Defant¹,

Schoolmann²

2021

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A particular consequence of the famous Carleson-Hunt theorem is that the Taylor series expansions of bounded holomorphic functions on the open unit disk converge almost everywhere on the boundary, whereas on single points the convergence may fail. In contrast, there exists an ordinary Dirichlet series a n n −s , which on the open right half-plane [Re > 0] converges to a bounded, holomorphic function -but diverges at each point of the imaginary line, although its limit function extends continuously to the closed right half plane. Inspired by a result of M. Riesz, we study the boundary behavior of holomorphic functions f on the right half-plane which for some ℓ ≥ 0 satisfy the growth condition |f (s)| = O((1 + |s|) ℓ ) and are generated by some Riesz germ, i.e., there is a frequency λ = (λ n ) and a λ-Dirichlet series a n e −λns such that on some open subset of [Re > 0] and for some m ≥ 0 the function f coincides with the pointwise limit (as x → ∞) of so-called (λ, m)-Riesz means λn 0 . Our main results present criteria for pointwise and uniform Riesz summability of such functions on the boundary line [Re = 0], which includes conditions that are motivated by classics like the Dini-test or the principle of localization.

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Riesz means in Hardy spaces on Dirichlet groups

Defant

Schoolmann

2020

Math. Ann.

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Vector-valued general Dirichlet series

Cited by 9 publications

References 53 publications

Holomorphic functions of finite order generated by Dirichlet series

Holomorphic functions of finite order generated by Dirichlet series

Riesz summability on boundary lines of holomorphic functions generated by Dirichlet series

Riesz means in Hardy spaces on Dirichlet groups

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