arXiv subjects

Mieczysław Mastyło

Publications and source records attributed to Mieczysław Mastyło.

At least 19 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.

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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.

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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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arXiv2025

Maximal regularity estimates for the abstract Cauchy problems

In this work, we extend the Da Prato-Grisvard theory of maximal regularity estimates for sectorial operators in interpolation spaces. Specifically, for any generator $-A$ of an analytic semigroup on a Banach space $X$, we identify the interpolation spaces between $X$ and the domain $D_{A}$ of $A$ in which the part of $A$ satisfies certain maximal regularity estimates. We also establish several new results concerning both homogeneous and inhomogeneous $L^1$-maximal regularity estimates, extending and completing recent findings in the literature. These results are motivated not only by applications to problems in areas such as fluid mechanics but also by the intrinsic theoretical interest of the subject. In particular, we address the optimal choice of data spaces for the Cauchy problem associated with $A$, ensuring the existence of strong solutions with global-in-time control of their derivatives. This control is measured via the homogeneous parts of the interpolation norms in the spatial variable and weighted Lebesgue norms over the time interval. Furthermore, we characterize weighted $L^1$-estimates and establish their relationship with unweighted estimates. Additionally, we reformulate the characterization condition for $L^1$-maximal regularity due to Kalton and Portal in a priori terms that do not rely on semigroup operators. Finally, we introduce a new interpolation framework for $L^p$-maximal regularity estimates, where $p \in (1, \infty)$, within interpolation spaces generated by non-classical interpolation functors.

math.AP↗

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↗

arXiv2023

A Christ-Kiselev maximal theorem in quasi-Banach function lattices

A Christ-Kiselev maximal theorem is proved for linear operators between quasi-Banach function lattices satisfying certain lattice geometrical conditions. The result is further explored for weighted Lorentz spaces, classical Lorentz spaces, and Wiener amalgams of Lebesgue function and sequence spaces. Extensions are made to Köthe dual operators and to operators on interpolation spaces of quasi-Banach function lattices. Several applications to maximal Fourier operators are presented.

math.FA↗

arXiv2023

The Fourier transform in weighted rearrangement invariant spaces

It is shown that if the Fourier transform is a bounded map on a rearrangement-invariant space of functions on $\mathbb R^n$, modified by a weight, then the weight is bounded above and below and the space is equivalent to $L^2$. Also, if it is bounded from $L^p$ to $L^q$, each modified by the same weight, then the weight is bounded above and below and $1\le p=q'\le 2$. Applications prove the non-boundedness on these spaces of an operator related to the Schrödinger equation.

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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 $\boldsymbolλ(X,Y)$ 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↗

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.

math.FA↗

arXiv2020

A sharp variant of the Marcinkiewicz theorem with multipliers in Sobolev spaces of Lorentz type

Given a bounded measurable function $σ$ on $\mathbb{R}^n$, we let $T_σ$ be the operator obtained by multiplication on the Fourier transform by $σ$. Let $0<s_1\le s_2\le \cdots \le s_n<1$ and $ψ$ be a Schwartz function on the real line whose Fourier transform $\widehatψ$ is supported in $[-2,-1/2]\cup[1/2,2]$ and which satisfies $\sum_{j \in \mathbb{Z}} \widehatψ\left(2^{-j} ξ\right)=1$ for all $ξ\neq 0$. In this work we sharpen the known forms of the Marcinkiewicz multiplier theorem by finding an almost optimal function space with the property that, if the function \begin{equation*} (ξ_1,\dots, ξ_n)\mapsto \prod_{i=1}^n (I-\partial_i^2)^{\frac {s_i}2} \Big[ \prod_{i=1}^n \widehatψ(ξ_i) σ(2^{j_1}ξ_1,\dots , 2^{j_n}ξ_n)\Big] \end{equation*} belongs to it uniformly in $j_1,\dots , j_n \in \mathbb Z$, then $T_σ$ is bounded on $ {L}^p(\mathbb R^n)$ when $ |\frac{1}{p}-\frac{1}{2} | < s_1$ and $1<p<\infty$. In the case where $s_i\neq s_{i+1}$ for all $i$, it was proved in [Grafakos, Israel J. Math., to appear] that the Lorentz space $L ^{\frac{1}{s_1},1} (\mathbb{R}^n) $ is the function space sought. In this work we address the significantly more difficult general case when for certain indices $i$ we might have $s_i=s_{i+1}$. We obtain a version of the Marcinkiewicz multiplier theorem in which the space $L ^{\frac{1}{s_1},1}$ is replaced by an appropriate Lorentz space associated with a certain concave function related to the number of terms among $s_2,\dots , s_n$ that equal $s_1$. Our result is optimal up to an arbitrarily small power of the logarithm in the defining concave function of the Lorentz space.

math.CA↗

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 $\sup_{1\le j \leq N} \big |\sum_{1 \leq i \leq K}γ_i(\cdot) a_{i,j}\big|$ where $(a_{i,j})$ denotes a matrix of scalars and the $(γ_i)$ 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.

math.FA↗

arXiv2020

A unified approach to compatibility theorems on invertible interpolated operators

We prove the stability of isomorphisms between Banach spaces generated by interpolation methods introduced by Cwikel-Kalton-Milman-Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also the so-called $\pm$ or $G_1$ and $G_2$ methods defined by Peetre and Gustavsson-Peetre. This result is used to show the existence of solution of certain operator analytic equation. A by product of these results is a more general variant of the Albrecht-Müller result which states that the interpolated isomorphisms satisfy uniqueness-of-inverses between interpolation spaces. We show applications for positive operators between Calderón function lattices. We also derive connections between the spectrum of interpolated operators.

math.FA↗

arXiv2019

Interpolation of compact bilinear operators

We investigate the stability of compactness of bilinear operators acting on the product of interpolation of Banach spaces. We develop a general framework for such results and our method applies to abstract methods of interpolation in the senseof Aronszajn and Gagliardo. A key step is to show an one-sided bilinear interpolation theorem on compactness for bilinear operators on couples satisfying an approximation property. We show applications to general cases, including Peetre's method and the general real interpolation methods.

math.FA↗

arXiv2017

Bohr's phenomenon for functions on the Boolean cube

We study the asymptotic decay of the Fourier spectrum of real functions $f\colon \{-1,1\}^N \rightarrow \mathbb{R}$ in the spirit of Bohr's phenomenon from complex analysis. Every such function admits a canonical representation through its Fourier-Walsh expansion $f(x) = \sum_{S\subset \{1,\ldots,N\}}\widehat{f}(S) x^S \,,$ where $x^S = \prod_{k \in S} x_k$. Given a class $\mathcal{F}$ of functions on the Boolean cube $\{-1, 1\}^{N} $, the Boolean radius of $\mathcal{F}$ is defined to be the largest $ρ\geq 0$ such that $\sum_{S}{|\widehat{f}(S)| ρ^{|S|}} \leq \|f\|_{\infty}$ for every $f \in \mathcal{F}$. We give the precise asymptotic behaviour of the Boolean radius of several natural subclasses of functions on finite Boolean cubes, as e.g. the class of all real functions on $\{-1, 1\}^{N}$, the subclass made of all homogeneous functions or certain threshold functions. Compared with the classical complex situation subtle differences as well as striking parallels occur.

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