arXiv subjects

Martín Mansilla

Publications and source records attributed to Martín Mansilla.

12 recordsLinked to original sources

arXiv2026

Projection Constants of Polynomial Spaces via Representation Theory

We develop a representation-theoretic framework for computing projection constants of finite-dimensional subspaces of $C(K)$, where $K$ is a compact homogeneous $G$-space. Within this framework, we characterize the finite-dimensional $G$-invariant subspaces for which the $G$-equivariant projection is unique. These are precisely the finite orthogonal sums of full isotypic components in the Peter--Weyl decomposition of $L_2(K)$. For such subspaces, averaging shows that this unique $G$-equivariant projection is minimal; it is the restriction to $C(K)$ of the $L_2(K)$-orthogonal projection. Consequently, their projection constants are given by the $L_1$-norm of an explicit reproducing-kernel slice. We apply this method to spaces of $d$-homogeneous polynomials in high dimension. The examples range from Fourier analysis on the torus, through spherical harmonic analysis on real and complex Euclidean spheres, to genuinely noncommutative harmonic analysis on the unitary group, where representation theory becomes indispensable. For all these families, we obtain precise high-dimensional asymptotics at the square-root-of-dimension scale.

math.FA↗

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.

math.FA↗

arXiv2026

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.

math.FA↗

arXiv2026

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.

math.FA↗

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.

math.FA↗

arXiv2024

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.

math.FA↗

arXiv2024

Projection constants for spaces of Dirichlet polynomials

Given a frequency sequence $ω=(ω_n)$ and a finite subset $J \subset \mathbb{N}$, we study the space $\mathcal{H}_{\infty}^{J}(ω)$ of all Dirichlet polynomials $D(s) := \sum_{n \in J} a_n e^{-ω_n s}, \, s \in \mathbb{C}$. The main aim is to prove asymptotically correct estimates for the projection constant $\boldsymbolλ\big(\mathcal{H}_\infty^{J}(ω) \big)$ of the finite dimensional Banach space $\mathcal{H}_\infty^{J}(ω)$ equipped with the norm $\|D\|= \sup_{\text{Re}\,s>0} |D(s)|$. Based on harmonic analysis on $ω$-Dirichlet groups, we prove the formula $ \boldsymbolλ\big(\mathcal{H}_\infty^{J}(ω) \big) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T \Big|\sum_{n \in J} e^{-iω_n t}\Big|\,dt\,, $ and apply it to various concrete frequencies $ω$ 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 $ \boldsymbolλ\big(\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)\big) \sim \sqrt{x}/(\log \log x)^{\frac{1}{4}}. $

math.FA↗

arXiv2024

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 $ \boldsymbolλ\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} \boldsymbolλ\big(\mathcal S_1(n)\big)/n = \sqrtπ/2\,. $

math.FA↗

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 $\boldsymbolλ(X,Y)$ of a subspace $X$ of a Banach $Y$ is the smallest norm among all possible projections on $Y$ onto $X$, and the projection constant $\boldsymbolλ(X)$ 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 $f(g)=\sum_{γ\in E} \hat{f}(γ) γ(g)$ defined on a compact topological group $G$, which have Fourier coefficients $\hat{f}(γ)$ supported in a finite set $E$ of characters; or analytic polynomials $P(z)=\sum_{α\in J}c_α(P)\,z^α$, which are defined on a Banach space $X_n = (\mathbb{C}^n, \|\cdot\|)$ and have monomial coefficients $c_α(P)$ 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.

math.FA↗

arXiv2019

Monomial convergence on $\ell_r$

For $1 < r \le 2$, we study the set of monomial convergence for spaces of holomorphic functions over $\ell_r$. For $ H_b(\ell_r)$, the space of entire functions of bounded type in $\ell_r$, we prove that $\mbox{mon} H_b(\ell_r)$ is exactly the Marcinkiewicz sequence space $m_{Ψ_r}$ where the symbol $Ψ_r$ is given by $Ψ_r(n) := \log(n + 1)^{1 - \frac{1}{r}}$ for $n \in \mathbb N_0$. For the space of $m$-homogeneous polynomials on $\ell_r$, we prove that the set of monomial convergence $\mbox{mon} \mathcal P (^m \ell_r)$ contains the sequence space $\ell_{q}$ where $q=(mr')'$. Moreover, we show that for any $q\leq s<\infty$, the Lorentz sequence space $\ell_{q,s}$ lies in $\mbox{mon} \mathcal P (^m \ell_r)$, provided that $m$ is large enough. We apply our results to make an advance in the description of the set of monomial convergence of $H_{\infty}(B_{\ell_r})$ (the space of bounded holomorphic on the unit ball of $\ell_r$). As a byproduct we close the gap on certain estimates related with the \emph{mixed} unconditionality constant for spaces of polynomials over classical sequence spaces.

math.CV↗

arXiv2018

The sup-norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials

Let $A_{p,r}^m(n)$ be the best constant that fulfills the following inequality: for every $m$-homogeneous polynomial $P(z) = \sum_{|α|=m} a_α z^α$ in $n$ complex variables, $$\big( \sum_{|α|=m} |a_α|^{r} \big)^{1/r} \leq A_{p,r}^m(n) \sup_{z \in B_{\ell_p^n}} \big|P(z) \big| .$$ For every degree $m$, and a wide range of values of $ p,r \in [1,\infty]$ (including any $r$ in the case $p \in [1,2]$, and any $r$ and $p$ for the 2-homogeneous case), we give the correct asymptotic behavior of these constants as $n$ (the number of variables) tends to infinity. Remarkably, in many cases, extremal polynomials for these inequalities are not (as traditionally expected) found using classical random unimodular polynomials, and special combinatorial configurations of monomials are needed. Namely, we show that Steiner polynomials (i.e., $m$-homogeneous polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems), do the work for certain range of values of $p,r$. As a byproduct, we present some applications of these estimates to the interpolation of tensor products of Banach spaces, to the study of (mixed) unconditionality in spaces of polynomials and to the multivariable von Neumann's inequality.

math.FA↗

arXiv2017

Mixed Bohr radius in several variables

Let $K(B_{\ell_p^n},B_{\ell_q^n}) $ be the $n$-dimensional $(p,q)$-Bohr radius for holomorphic functions on $\mathbb C^n$. That is, $K(B_{\ell_p^n},B_{\ell_q^n}) $ denotes the greatest constant $r\geq 0$ such that for every entire function $f(z)=\sum_α c_α z^α$ in $n$-complex variables, we have the following (mixed) Bohr-type inequality $$\sup_{z \in r \cdot B_{\ell_q^n}} \sum_α | c_α z^α | \leq \sup_{z \in B_{\ell_p^n}} | f(z) |,$$ where $B_{\ell_r^n}$ denotes the closed unit ball of the $n$-dimensional sequence space $\ell_r^n$. For every $1 \leq p, q \leq \infty$, we exhibit the exact asymptotic growth of the $(p,q)$-Bohr radius as $n$ (the number of variables) goes to infinity.

math.CV↗