arXiv subjects

Melisa Scotti

Publications and source records attributed to Melisa Scotti.

4 recordsLinked to original sources

arXiv2026

Dilation systems with two generators and the optimal Riesz bounds of a basis of ReLU networks

For two $1$-periodic functions $ψ_1,ψ_2\in L^2(0,1)$ with mean zero we characterize when $\{1\}\cup\{ψ_1(α\cdot x),ψ_2(α\cdot x)\}_{α\in\mathbb{Z}^d,\,α\succ0}$ is a Riesz basis of $L^2([0,1]^d)$, in terms of a $2\times2$ matrix of Dirichlet series, and we compute its optimal bounds. The tool is a $2\times2$ matrix version of the theorem of Hedenmalm, Lindqvist and Seip for systems of dilations with two generators, proved through the reproducing kernel of the Hardy space of the infinite polydisk. In particular, we get the optimal bounds $2/3$ and $3/2$ for $ψ_1,ψ_2$ the triangle waves considered by Daubechies, DeVore, Foucart, Hanin and Petrova in $L^2(0,1)$, and the extended version in $L^2([0,1]^d)$ defined by Schneider and Vybíral.

math.CA↗

arXiv2025

Characterization of multipliers on vector-valued Hardy spaces

This work characterizes the multipliers on vector-valued Hardy spaces over the infinite polydisk and the infinite polytorus, as well as in the context of Dirichlet series. Unlike the scalar-valued setting, where these frameworks are completely analogous reformulations of one another, there are significant differences in the vector-valued context. We prove that while the space of multipliers on the infinite polydisk is $H_\infty(\mathbb{D}^\infty_2, B(X))$, the situation on the infinite polytorus is distinct; assuming $X$ is separable, the multiplier space can be identified as $H_\infty^{sot}(\mathbb{T}^\infty, B(X))$, consisting of essentially bounded SOT-measurable functions. These spaces coincide when $X$ possesses the analytic Radon-Nikodym property. Finally, we extend these results to the associated Hardy spaces of Dirichlet series, $\mathcal{H}_p^+(X)$ and $\mathcal{H}_p(X)$, providing characterizations for their respective multiplier spaces.

math.FA↗

arXiv2021

Splitting the Riesz basis condition for systems of dilated functions]{Splitting the Riesz basis condition for systems of dilated functions through Dirichlet series

Inspired by the work of Hedenmalm, Lindqvist and Seip, we consider different properties of dilations systems of a fixed function $φ\in L^2(0,1)$. More precisely, we study when the system $\{φ(nx)\}_n$ is a Bessel sequence, a Riesz sequence, or it satisfies the lower frame bound. We are able to characterize these properties in terms of multipliers of the Hardy space $\mathcal{H}^2$ of Dirichtet series and, also, in terms of Hardy spaces on the infinite polytorus. We also address the multivariate case.

math.FA↗

arXiv2018

Random unconditional convergence of vector-valued Dirichlet series

We study random unconditionality of Dirichlet series in vector-valued Hardy spaces $\mathcal H_p(X)$. It is shown that a Banach space $X$ has type 2 (respectively, cotype 2) if and only if for every choice $(x_n)_n\subset X$ it follows that $(x_n n^{-s})_n$ is Random unconditionally convergent (respectively, divergent) in $\mathcal H_2(X)$. The analogous question on $\mathcal H_p(X)$ spaces for $p\neq2$ is also explored. We also provide explicit examples exhibiting the differences between the unconditionality of $(x_n n^{-s})_n$ in $\mathcal H_p(X)$ and that of $(x_n z^n)_n$ in $H_p(X)$.

math.FA↗