Hilbert spaces of Dirichlet series and their multipliers

mathrm{c}}$Carthy, John M$^{

doi:10.1090/s0002-9947-03-03452-4

Cited by 32 publications

(30 citation statements)

References 11 publications

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“…(See [8,21,23,29] for further results on functions spaces of Dirichlet series.) Such precise results are perhaps surprising in view of a deep feature of the theory, which dates back to H. Bohr [4].…”

Section: Introductionmentioning

confidence: 99%

Local properties of Hilbert spaces of Dirichlet series

Olsen¹

2011

Journal of Functional Analysis

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We show that the asymptotic behavior of the partial sums of a sequence of positive numbers determine the local behavior of the Hilbert space of Dirichlet series defined using these as weights. This extends results recently obtained describing the local behavior of Dirichlet series with square summable coefficients in terms of local integrability, boundary behavior, Carleson measures and interpolating sequences. As these spaces can be identified with functions spaces on the infinite-dimensional polydisk, this gives new results on the Dirichlet and Bergman spaces on the infinite dimensional polydisk, as well as the scale of Besov-Sobolev spaces containing the Drury-Arveson space on the infinite dimensional unit ball. We use both techniques from the theory of sampling in Paley-Wiener spaces, and classical results from analytic number theory.We study the connection between the asymptotic behavior in terms of the inequalities (1) for general sequences (w n ) n∈N of non-negative numbers, and local 1 arXiv:1011.3370v1 [math.CV]

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Section: Introductionmentioning

confidence: 99%

Local properties of Hilbert spaces of Dirichlet series

Olsen¹

2011

Journal of Functional Analysis

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“…The solution to the [0, 1]-moment problem (which is a simple consequence of [16, Proposition 1.9]) allows us to answer Question 1.1 (see First proof of Theorem 3.1). Note that a particular case of the [0, ∞)moment problem appears in [11,Equation (1.3)] in the study of the multiplier algebra of certain Hilbert spaces of the Dirichlet series (see Example 2.1(a)). This problem also appears in the study of bounded Helson matrices (see [12]).…”

Section: Hausdorff Log-moment Sequencesmentioning

confidence: 99%

“…In case j 2, representing measures of Hausdorff log-moment sequences are not necessarily finite (cf. [11,Section 1]). (a) For a real number α < 0, the sequence {(log(n)) α } n 2 is a Hausdorff log-moment sequence with representing measure µ α equal to…”

Section: Hausdorff Log-moment Sequencesmentioning

confidence: 99%

Dirichlet polynomials and a moment problem

Chavan¹,

Sahu²

2021

Preprint

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Consider a linear functional L defined on the space D[s] of Dirichlet polynomials with real coefficients and the set D+[s] of nonnegative elements in D[s]. An analogue of the Riesz-Haviland theorem in this context asks: What are all D+[s]-positive linear functionals L, which are moment functionals? Since the space D[s], when considered as a subspace of C([0, ∞), R), fails to be an adapted space in the sense of Choquet, the general form of Riesz-Haviland theorem is not applicable in this situation. In an attempt to answer the forgoing question, we arrive at the notion of a moment sequence, which we call the Hausdorff log-moment sequence. Apart from an analogue of the Riesz-Haviland theorem, we show that any Hausdorff log-moment sequence is a linear combination of {1, 0, . . . , } and {f (log(n)} n 1 for a completely monotone function f : [0, ∞) → [0, ∞). Moreover, such an f is uniquely determined by the sequence in question.

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“…The space H corresponds to H p for p = 2. In [19], McCarthy considered the weighted Hilbert spaces of Dirichlet series…”

Section: Introductionmentioning

confidence: 99%

“…This identification was a key step in solving Beurling's question on complete sequences in L 2 (0, 1). It is noticeable that for all the spaces H p , H α , A p μ , B p , D α the multiplier algebra is also the algebra H ∞ ; [6, Theorem 7], [19,Theorem 1.11], [2, Theorem 10.1 and Theorem 11.21], [3,Theorem 3].…”

Section: Introductionmentioning

confidence: 99%