arXiv subjects

Andreas Defant

Publications and source records attributed to Andreas Defant.

At least 19 recordsLinked to original sources

arXiv2026

Support-Sensitive Bohnenblust-Hille Inequalities and Local Invariants on Hamming Schemes

We investigate local invariants and geometric phenomena for polynomial spaces of low degree on the $q$-ary Hamming scheme $C_q^N$, where $C_q$ denotes the cyclic group of order $q$. Our main analytic tool is a support-sensitive Bohnenblust--Hille inequality for spherical polynomial spaces, showing that the relevant complexity parameter is the support size of the monomials rather than their total degree. Equivalently, in the corresponding toroidal formulation, this leads to estimates for polynomials whose coordinate degrees are bounded by $q-1$, while the growth of the constants is governed by the interaction order of the variables. These inequalities yield applications to the learning theory of spherical low-level functions and also provide the basis for dimension-free comparisons between several classical local invariants, including Sidon constants, unconditional basis constants, and Gordon--Lewis constants. As a consequence, we obtain sharp asymptotic estimates for these invariants in the spherical setting, with analogous comparison and asymptotic results for homogeneous and tetrahedral polynomial spaces. We also study projection constants and the associated reproducing kernels. In the spherical case, suitably normalized Krawtchouk polynomials converge to Hermite polynomials under central-limit scaling, leading to explicit Gaussian limits and sharp asymptotic formulas. By contrast, in the homogeneous and tetrahedral settings a dichotomy appears between the Boolean case and the regime $q\ge3$, where the limiting behaviour is governed by moments of a circular complex Gaussian.

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arXiv2025

Local constants and Bohr's phenomenon for Banach spaces of analytic polynomials

The primary aim of this work is to develop methods that provide new insights into the relationships between fundamental constants in Banach space theory--specifically, the projection constant, the unconditional basis constant and the Gordon-Lewis constant--for the Banach space $\mathcal{P}_J(X_n)$ of multivariate analytic polynomials. This class consists of all polynomials whose monomial coefficients vanish outside the set of multi-indices $J$, and it is equipped with the supremum norm on the unit sphere of the finite-dimensional Banach space $X_n = (\mathbb{C}^n, \|\cdot\|)$. We establish a~general framework for proving quantitative results on the asymptotic optimal behavior of these constants, which depend on both the dimension of the space and the degree of the polynomials. Using the tools developed, we derive asymptotic estimates of the Bohr radius for general Banach sequence lattices. Additionally, we apply our results to the asymptotic study of local constants and the Bohr radius within finite-dimensional Lorentz sequence spaces, which requires a~refined analysis of the combinatorial structure of the associated index sets. As a consequence, we obtain optimal results across a broad range of parameters.

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arXiv2024

Ryll-Wojtaszczyk Formulas for bihomogeneous polynomials on the sphere

We investigate projection constants for spaces of bihomogeneous harmonic and bihomogeneous polynomials on the unit sphere in finite-dimensional complex Hilbert spaces. Using averaging techniques, we demonstrate that the minimal norm projection aligns with the natural orthogonal projection. This result enables us to establish a connection between these constants and weighted \linebreak $L_1$-norms of specific Jacobi polynomials. Consequently, we derive explicit bounds, provide practical expressions for computation, and present asymptotically sharp estimates for these constants. Our findings extend the classical Ryll and Wojtaszczyk formula for the projection constant of homogeneous polynomials in finite-dimensional complex Hilbert spaces to the bihomogeneous setting.

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arXiv2024

Minimal projections onto spaces of polynomials on real euclidean spheres

We investigate projection constants within classes of multivariate polynomials over finite-dimensional real Hilbert spaces. Specifically, we consider the projection constant for spaces of spherical harmonics and spaces of homogeneous polynomials as well as for spaces of polynomials of finite degree on the unit sphere. We establish a connection between these quantities and certain weighted $L_1$-norms of specific Jacobi polynomials. As a consequence, we present exact formulas, computable expressions and asymptotically accurate estimates for them. The real case we address is considerably more nuanced than its complex counterpart.

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arXiv2024

Hypercontractivity and strips of convergence in Hardy spaces of general Dirichlet series

For a general Dirichlet series $\sum a_n e^{-\lambda_n s}$ with frequency $\lambda=(\lambda_n)_n$, we study how horizontal translation (i.e. convolution with a Poisson kernel) improves its integrability properties. We characterize hypercontractive frequencies in terms of their additive structure answering some questions posed by Bayart. We also provide sharp bounds for the strips $S_p(\lambda)$ that encode the minimum translation necessary for series in the Hardy space $\mathcal{H}_p(\lambda)$ to have absolutely convergent coefficients.

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arXiv2023

Projection constants for spaces of Dirichlet polynomials

Given a frequency sequence $\omega=(\omega_n)$ and a finite subset $J \subset \mathbb{N}$, we study the space $\mathcal{H}_{\infty}^{J}(\omega)$ of all Dirichlet polynomials $D(s) := \sum_{n \in J} a_n e^{-\omega_n s}, \, s \in \mathbb{C}$. The main aim is to prove asymptotically correct estimates for the projection constant $\boldsymbol{\lambda}\big(\mathcal{H}_\infty^{J}(\omega) \big)$ of the finite dimensional Banach space $\mathcal{H}_\infty^{J}(\omega)$ equipped with the norm $\|D\|= \sup_{\text{Re}\,s>0} |D(s)|$. Based on harmonic analysis on $\omega$-Dirichlet groups, we prove the formula $ \boldsymbol{\lambda}\big(\mathcal{H}_\infty^{J}(\omega) \big) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T \Big|\sum_{n \in J} e^{-i\omega_n t}\Big|\,dt\,, $ and apply it to various concrete frequencies $\omega$ and index sets $J$. To see an example, combining with a recent deep result of Harper from probabilistic analytic number theory, we for the space $\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)$ of all ordinary Dirichlet polynomials $D(s) = \sum_{n \leq x} a_n n^{-s}$ of length $x$ show the asymptotically correct order $ \boldsymbol{\lambda}\big(\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)\big) \sim \sqrt{x}/(\log \log x)^{\frac{1}{4}}. $

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arXiv2023

Asymptotic insights for projection, Gordon-Lewis and Sidon constants in Boolean cube function spaces

The main aim of this work is to study important local Banach space constants for Boolean cube function spaces. Specifically, we focus on $\mathcal{B}_{\mathcal{S}}^N$, the finite-dimensional Banach space of all real-valued functions defined on the $N$-dimensional Boolean cube $\{-1, +1\}^N$ that have Fourier--Walsh expansions supported on a fixed~family $\mathcal{S}$ of subsets of $\{1, \ldots, N\}$. Our investigation centers on the projection, Sidon and Gordon--Lewis constants of this function space. We combine tools from different areas to derive exact formulas and asymptotic estimates of these parameters for special types of families $\mathcal{S}$ depending on the dimension $N$ of the Boolean cube and other complexity characteristics of the support set $\mathcal{S}$. Using local Banach space theory, we establish the intimate relationship among these three important constants.

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arXiv2023

The projection constant for the trace class

We study the projection constant of the space of operators on $n$-dimensional Hilbert spaces, with the trace norm, $\mathcal S_1(n)$. We show an integral formula for the projection constant of $\mathcal S_1(n)$; namely $ \boldsymbol{\lambda}\big(\mathcal S_1(n)\big) = n \int_{\mathcal U_n} \vert \text{tr}(U) \vert \,dU \,, $ where the integration is with respect to the Haar probability measure on the group $\mathcal U_n$ of unitary operators. Using a probabilistic approach, we derive the limit formula $ \lim_{n\to \infty} \boldsymbol{\lambda}\big(\mathcal S_1(n)\big)/n = \sqrt{\pi}/2\,. $

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arXiv2022

Projection constants for spaces of multivariate polynomials

The general problem we address is to develop new methods in the study of projection constants of Banach spaces of multivariate polynomials. The relative projection constant $\boldsymbol{\lambda}(X,Y)$ of a subspace $X$ of a Banach $Y$ is the smallest norm among all possible projections on $Y$ onto $X$, and the projection constant $\boldsymbol{\lambda}(X)$ is the supremum of all relative projection constants of $X$ taken with respect to all possible super spaces $Y$. This is one of the most significant notions of modern Banach space theory and has been intensively studied since the birth of abstract operator theory. We focus on projection constants of Banach spaces of multivariate polynomials formed either by trigonometric polynomials $f(g)=\sum_{\gamma \in E} \hat{f}(\gamma) \gamma(g)$ defined on a compact topological group $G$, which have Fourier coefficients $\hat{f}(\gamma)$ supported in a finite set $E$ of characters; or analytic polynomials $P(z)=\sum_{\alpha\in J}c_\alpha(P)\,z^\alpha$, which are defined on a Banach space $X_n = (\mathbb{C}^n, \|\cdot\|)$ and have monomial coefficients $c_\alpha(P)$ supported in a finite set $J \subset \mathbb{N}_0^n$ of multi indices. Depending on the underlying structure (of the group, Banach space or index set), the goal is to prove precise formulas or asymptotically optimal estimates. Our general setting is flexible enough to handle a wide variety of Banach spaces of polynomials, including analytic polynomials on polydiscs, Dirichlet polynomials on the complex plane, and polynomials on Boolean cubes $\{-1,+1\}^n$. Moreover, we get an explicit formula for the projection constant of the space of trace class operators. The methods developed here enable us to prove new estimates for important invariants such as the unconditional basis constant and the Gordon-Lewis constant for Banach spaces of multivariate polynomials.

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arXiv2021

Variants of a multiplier theorem of Kislyakov

We prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislaykov and the Kahane-Salem-Zygmund inequality. As a by-product we show various multiplier theorems for spaces of trigonometric polynomials on the $n$-dimensional torus $\mathbb{T}^n$ or Boolean cubes $\{-1,1\}^N$. Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application we show that various recent $\ell_1$-multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane-Salem-Zygmund inequality.

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arXiv2021

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

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, Bayart, Konyagin, and Queff\'elec constructed an example of an ordinary Dirichlet series $\sum a_n n^{-s}$, which on the open right half-plane $[Re >0]$ converges pointwise 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 $\ell \ge 0$ satisfy the growth condition $|f(s)| = O((1 + |s|)^\ell)$ and are generated by some Riesz germ, i.e., there is a frequency $\lambda = (\lambda_n)$ and a $\lambda$-Dirichlet series $\sum a_n e^{-\lambda_n s}$ such that on some open subset of $[Re >0]$ and for some $m \ge 0$ the function $f$ coincides with the pointwise limit (as $x \to \infty$) of so-called $(\lambda,m)$-Riesz means $\sum_{\lambda_n < x} a_n e^{-\lambda_n s}\big( 1-\frac{\lambda_n}{x}\big)^m ,\,x >0\,.$ 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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arXiv2021

Holomorphic functions of finite order generated by Dirichlet series

Given a frequency $\lambda = (\lambda_n)$ and $\ell \ge 0$, we introduce the scale of Banach spaces $H_{\infty,\ell}^{\lambda}[Re > 0]$ of holomorphic functions $f$ on the open right half-plane $[Re > 0]$, which satisfy $(A)$ the growth condition $|f(s)| = O((1 + |s|)^\ell)$, and $(B)$ have a Riesz germ, i.e. on some open subset and for some $m \ge 0$ the function $f$ coincides with the pointwise limit (as $x \to \infty$) of the so-called $(\lambda,m)$-Riesz means $\sum_{\lambda_n < x} a_n e^{-\lambda_n s}\big( 1-\frac{\lambda_n}{x}\big)^m ,\,x >0$ of some $\lambda$-Dirichlet series $\sum a_n e^{-\lambda_n s}$. Reformulated in our terminology, an important result of M. Riesz shows that in this case the function $f$ for every $k >\ell$ is the pointwise limit of the $(\lambda,k)$-Riesz means of $D$ on $[Re > 0]$. Our main contribution is an extension -- showing that 'after translation' every bounded set in $H_{\infty,\ell}^{\lambda}[Re > 0]$ is uniformly approximable by all its $(\lambda,k)$-Riesz means of order $k>\ell$. 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 spaces $H_{\infty,\ell}^{\lambda}[Re > 0]$. One of the many consequences is that $H_{\infty,\ell}^{\lambda}[Re > 0]$ basically consists of those holomorphic functions on $[Re >0]$, which have a Riesz germ and are of finite uniform order $\ell$ on $[Re >0]$. 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.

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arXiv2020

Fr\'echet spaces of general Dirichlet series

Inspired by a recent article on Fr\'echet spaces of ordinary Dirichlet series $\sum a_n n^{-s}$ due to J.~Bonet, we study topological and geometrical properties of certain scales of Fr\'echet spaces of general Dirichlet spaces $\sum a_n e^{-\lambda_n s}$. More precisely, fixing a frequency $\lambda = (\lambda_n)$, we focus on the Fr\'echet space of $\lambda$-Dirichlet series which have limit functions bounded on all half planes strictly smaller than the right half plane $[\mathrm{Re} >0]$. We develop an abstract setting of pre-Fr\'echet spaces of $\lambda$-Dirichlet series generated by certain admissible normed spaces of $\lambda$-Dirichlet series and the abscissas of convergence they generate, which allows also to define Fr\'echet spaces of $\lambda$-Dirichlet series for which $a_n e^{-\lambda_n/k}$ for each $k$ equals the Fourier coefficients of a function on an appropriate $\lambda$-Dirichlet group.

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arXiv2020

Subgaussian Kahane-Salem-Zygmund inequalities in Banach spaces

The main aim of this work is to give a general approach to the celebrated Kahane-Salem-Zygmund inequalities. We prove estimates for exponential Orlicz norms of averages $\sup_{1\le j \leq N} \big |\sum_{1 \leq i \leq K}\gamma_i(\cdot) a_{i,j}\big|$ where $(a_{i,j})$ denotes a matrix of scalars and the $(\gamma_i)$ a sequence of real or complex subgaussian random variables. Lifting these inequalities to finite dimensional Banach spaces, we get novel Kahane-Salem-Zygmund type inequalities -- in particular, for spaces of subgaussian random polynomials and multilinear forms on finite dimensional Banach spaces as well as subgaussian random Dirichlet polynomials. Finally, we use abstract interpolation theory to widen our approach considerably.

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arXiv2019

Variants of a theorem of Helson on general Dirichlet series

A result of Helson on general Dirichlet series $\sum a_{n} e^{-\lambda_{n}s}$ states that, whenever $(a_{n})$ is $2$-summable and $\lambda=(\lambda_{n})$ satisfies a certain condition introduced by Bohr, then for almost all homomorphism $\omega \colon (\mathbb{R},+) \to \mathbb{T}$ the Dirichlet series $\sum a_{n} \omega(\lambda_{n})e^{-\lambda_{n}s}$ converges on the open right half plane $[Re>0]$. For ordinary Dirichlet series $\sum a_n n^{-s}$ Hedenmalm and Saksman related this result with the famous Carleson-Hunt theorem on pointwise convergence of Fourier series, and Bayart extended it within his theory of Hardy spaces $\mathcal{H}_p$ of such series. The aim here is to prove variants of Helson's theorem within our recent theory of Hardy spaces $\mathcal{H}_{p}(\lambda),\,1\le p \le \infty,$ of general Dirichlet series. To be more precise, in the reflexive case $1 < p < \infty$ we extend Helson's result to Dirichlet series in $\mathcal{H}_{p}(\lambda)$ without any further condition on the frequency $\lambda$, and in the non-reflexive case $p=1$ to the wider class of frequencies satisfying the so-called Landau condition (more general than Bohr's condition). In both cases we add relevant maximal inequalities. Finally, we give several applications to the structure theory of Hardy spaces of general Dirichlet series.

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arXiv2019

Riesz means in Hardy spaces on Dirichlet groups

Given a frequency $\lambda=(\lambda_n)$, we study when almost all vertical limits of a $\mathcal{H}_1$-Dirichlet series $\sum a_n e^{-\lambda_ns}$ are Riesz-summable almost everywhere on the imaginary axis. Equivalently, this means to investigate almost everywhere convergence of Fourier series of $H_1$-functions on so-called $\lambda$-Dirichlet groups, and as our main technical tool we need to invent a weak-type $(1, \infty)$ Hardy-Littlewood maximal operator for such groups. Applications are given to $H_1$-functions on the infinite dimensional torus $\mathbb{T}^\infty$, ordinary Dirichlet series $\sum a_n n^{-s}$, as well as bounded and holomorphic functions on the open right half plane, which are uniformly almost periodic on every vertical line.

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arXiv2019

Henry Helson meets other big shots -- A brief survey

A theorem of Henry Helson shows that for every ordinary Dirichlet series $\sum a_n n^{-s}$ with a square summable sequence $(a_n)$ of coefficients, almost all vertical limits $\sum a_n \chi(n) n^{-s}$, where $\chi: \mathbb{N} \to \mathbb{T}$ is a completely multiplicative arithmetic function, converge on the right half-plane. We survey on recent improvements and extensions of this result within Hardy spaces of Dirichlet series -- relating it with some classical work of Bohr, Banach, Carleson-Hunt, Ces\`{a}ro, Hardy-Littlewood, Hardy-Riesz, Menchoff-Rademacher, and Riemann.

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arXiv2019

Hardy spaces of general Dirichlet series - a survey

The main purpose of this article is to survey on some key elements of a recent $\mathcal{H}_p$-theory of general Dirichlet series $\sum a_n e^{-\lambda_{n}s}$, which was mainly inspired by the work of Bayart and Helson on ordinary Dirichlet series $\sum a_n n^{-s}$. In view of an ingenious identification of Bohr, the $\mathcal{H}_p$-theory of ordinary Dirichlet series can be seen as a sub-theory of Fourier analysis on the infinite dimensional torus $\mathbb{T}^\infty$. Extending these ideas, the $\mathcal{H}_p$-theory of $\lambda$-Dirichlet series is build as a sub-theory of Fourier analysis on what we call $\lambda$-Dirichlet groups. A number of problems is added.

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