arXiv math/94042081994-04-12math.FA

Ideal norms and trigonometric orthonormal systems

Joerg Wenzel

英文摘要：

In this article, we characterize the $UMD$--property of a Banach space $X$ by ideal norms associated with trigonometric orthonormal systems. The asymptotic behavior of that numerical parameters can be used to decide whether or not $X$ is a $UMD$--space. Moreover, in the negative case, we obtain a measure that shows how far $X$ is from being a $UMD$--space. The main result is, that all described parameters are equivalent also in the quantitative setting.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A characterization for complex symmetric Toeplitz operators

In this paper we use orthonormal basis for the Hardy space $H^{2}(\mathbb{T})$, formed by rational functions, to characterize complex symmetric Toeplitz operators on $H^{2}(\mathbb{T})$. As a result, we get examples of these operators whose symbols are non-trigonometric functions.

2022-11-24math.FA

On the equivalence between the sets of the trigonometric polynomials

In this paper we construct an injection from the linear space of trigonometric polynomials defined on $\mathbb{T}^d$ with bounded degrees with respect to each variable to a suitable linear subspace $L^1_E\subset L^1(\mathbb{T})$. We give such a quantitative condition on $L^1_E$ that this injection is a isomorphism of a Banach spaces equipped with $L^1$ norm and the norm of the isomorphism is independent on the dimension $d$.

2014-12-16math.CAmath.FA

Orthonormal pairs of operators

We consider pairs of operators $A,B\in B(H)$, where $H$ is a Hilbert space, such that there exist a linear isometry $f$ from the span of $\{A,B\}$ into $\mathbb{C}^2$ mapping $A,B$ into orthonormal vectors. We prove some necessary conditions for the existence of such an $f$ and determine all such pairs among commuting normal operators. Then we characterize all such pairs $A,B$ (in fact, we consider general sets instead of just pairs) under the additional requirement that $f$ is a complete isometry, when $H$ carries the column (or the row) operator space structure. We also metrically characteri…

2022-07-05math.FA

Unconditionality with respect to orthonormal systems in noncommutative $L_2$ spaces

Orthonormal systems in commutative $L_2$ spaces can be used to classify Banach spaces. When the system is complete and satisfies certain norm condition the unconditionality with respect to the system characterizes Hilbert spaces. As a noncommutative analogue we introduce the notion of unconditionality of operator spaces with respect to orthonormal systems in noncommutative $L_2$ spaces and show that the unconditionality characterizes operator Hilbert spaces when the system is complete and satisfy certain norm condition. The proof of the main result heavily depends on free probabilistic tools s…

2006-10-07math.FAmath.OA

Stable decomposition of homogeneous Mixed-norm Triebel-Lizorkin spaces

We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel-Lizorkin spaces in an anisotropic setting on $\bR^d$. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel-Lizorkin spaces. In the second part of the paper we study nonlinear $m$-term nonlinear approximation with the constructed basis in the mixed-norm setting, where the beh…

2022-06-08math.FA

Majorizing measures and proportional subsets of bounded orthonormal systems

In this article we prove that for any orthonormal system $(\vphi_j)_{j=1}^n \subset L_2$ that is bounded in $L_{\infty}$, and any $1 < k <n$, there exists a subset $I$ of cardinality greater than $n-k$ such that on $\spa\{\vphi_i\}_{i \in I}$, the $L_1$ norm and the $L_2$ norm are equivalent up to a factor $μ(\log μ)^{5/2}$, where $μ= \sqrt{n/k} \sqrt{\log k}$. The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures.

2008-01-23math.FAmath.PR

更多相关论文收起更多论文

The maximum modulus of a trigonometric trinomial

Let Lambda be a set of three integers and let C_Lambda be the space of 2pi-periodic functions with spectrum in Lambda endowed with the maximum modulus norm. We isolate the maximum modulus points x of trigonometric trinomials T in C_Lambda and prove that x is unique unless |T| has an axis of symmetry. This permits to compute the exposed and the extreme points of the unit ball of C_Lambda, to describe how the maximum modulus of T varies with respect to the arguments of its Fourier coefficients and to compute the norm of unimodular relative Fourier multipliers on C_Lambda. We obtain in particular…

2007-03-08math.CAmath.FA

Type and cotype with respect to arbitrary orthonormal systems

Let $\on_{k \in \nz}$ be an orthonormal system on some $σ$-finite measure space $(\Om,p)$. We study the notion of cotype with respect to $Φ$ for an operator $T$ between two Banach spaces $X$ and $Y$, defined by $\fco T := \inf$ $c$ such that \[ \Tfmm \pl \le \pl c \pll \gmm \hspace{.7cm}\mbox{for all}\hspace{.7cm} (x_k)\subset X \pl,\] where $(g_k)_{k\in \nz}$ is a sequence of independent and normalized gaussian variables. It is shown that this $Φ$-cotype coincides with the usual notion of cotype $2$ iff \linebreak $\fco {I_{\lin}} \sim \sqrt{\frac{n}{\log (n+1)}}$ uniformly in $n$ iff there i…

1994-01-04math.FA

双曲交叉三角多项式一致范数的采样离散化

Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials

University of Alberta（阿尔伯塔大学）；University of Manitoba（曼尼托巴大学）

研究双曲交叉三角多项式一致范数的采样离散化，构造显式赋范集并证明随机采样达到最优指数，同时给出赋范集大小的下界，排除二次界。

2026-10-07cs.NAmath.CAmath.FA

Approximation by nonlinear Fourier Basis in generalized Holder spaces

In this paper, the value of deviation of a function f from its nth generalized de la Vallee-Poussin mean with respect to the nonlinear trigonometric system is estimated for the classes of 2 pi-periodic functions in the uniform norm and in the generalized Holder norm .

2019-11-08math.FA

与傅里叶余弦-正弦变换和Kontorovich-Lebedev变换相关的三角加权广义卷积算子

Trigonometric weighted generalized convolution operator associated with Fourier cosine-sine and Kontorovich-Lebedev transformations

本研究引入带三角权重γ=sin y、关联傅里叶余弦-正弦与Kontorovich-Lebedev变换的广义卷积算子，证明其在双参数Lebesgue空间的有界性，给出加权L_p范数估计并得到相关卷积积分微分方程的L_1可解条件。

2024-04-12math.CAmath.FA

Hilbert空间中框架与基的对称逼近

Symmetric approximation of frames and bases in Hilbert spaces

本文研究Hilbert空间中框架和基的对称逼近问题，利用算子(P-|F|)的Hilbert-Schmidt性质刻画归一化紧框架和标准正交基的二次最近逼近的存在性与唯一性，结果可应用于小波理论。

1998-12-08math.FA

最佳$m$-项三角逼近在加权Wiener空间中的研究及其应用

Best $m$-term trigonometric approximation in weighted Wiener spaces and applications

本文研究了加权Wiener空间中的最佳$m$-项三角逼近，并通过压缩感知算法改进了多变量函数空间的恢复界。

2025-11-19cs.NAmath.FAmath.NA

二维空间中标准正交族的半经典Moser-Trudinger型不等式

Semiclassical Moser--Trudinger type inequalities for orthonormal families in $\mathbb{R}^2$

Masaryk University（马萨里克大学）

本文针对二维空间中H¹标准正交族，证明了带最优系数4π的全空间端点Moser-Trudinger不等式，通过热半群改进Rumin密度方法处理相关问题，还揭示了不等式中秩依赖项N-1的不可或缺性。

2026-10-14math-phmath.FAmath.MP

On some classical problems concerning $L_{\infty}$-extremal polynomials with constraints

First we consider the following problem which dates back to Chebyshev, Zolotarev and Achieser: among all trigonometric polynomials with given leading coefficients $a_0,...,a_l,$ $b_0,...,b_l \in \mathbb R$ find that one with least maximum norm on $[0,2 π].$ We show that the minimal polynomial is on $[0,2 π]$ asymptotically equal to a Blaschke product times a constant where the constant is the greatest singular value of the Hankel matrix associated with the $τ_j = a_j + i b_j.$ As a special case corresponding statements for algebraic polynomials follow. Finally the minimal norm of certain linea…

2010-01-04math.CAmath.FA

Polytopes of eigensteps of finite equal norm tight frames

Hilbert space frames generalize orthonormal bases to allow redundancy in representations of vectors while keeping good reconstruction properties. A frame comes with an associated frame operator encoding essential properties of the frame. We study a polytope that arises in an algorithm for constructing all finite frames with given lengths of frame vectors and spectrum of the frame operator, which is a Gelfand-Tsetlin polytope. For equal norm tight frames, we give a non-redundant description of the polytope in terms of equations and inequalities. From this we obtain the dimension and number of f…

2016-06-19math.COmath.FAmath.MG