Journal of Approximation Theory最新文献

英文中文

Sharp iteration asymptotics for transfer operators induced by greedy β-expansions 由贪婪β-展开诱导的转移算子的尖锐迭代渐近性

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-09-03 DOI: 10.1016/j.jat.2025.106234

Horia D. Cornean, Kasper S. Sørensen

We consider base-

β

expansions of Parry’s type, where

a_{0} \geq a_{1} \geq 1

are integers and

a_{0} < β < a_{0} + 1

is the positive solution to

β^{2} = a_{0} β + a_{1}

(the golden ratio corresponds to

a_{0} = a_{1} = 1

). The map

x \mapsto β x - ⌊ β x ⌋

induces a discrete dynamical system on the interval

[0, 1)

and we study its associated transfer (Perron–Frobenius) operator

P

. Our main result can be roughly summarized as follows: we explicitly construct two piecewise affine functions

u

and

v

with

P u = u

and

P v = β^{- 1} v

such that for every sufficiently smooth

F

which is supported in

[0, 1]

and satisfies

\int_{0}^{1} F d x = 1

, we have

P^{k} F = u + β^{- k} (F (1) - F (0)) v + o (β^{- k})

L^{\infty}

. This is also compared with the case of integer bases, where more refined asymptotic formulas are possible.

我们考虑Parry型的碱-β展开式，其中a0≥a1≥1是整数，a0<β<；a0+1是β2=a0β+a1的正解（黄金比例对应于a0=a1=1）。映射x∈βx−⌊βx⌋在区间[0,1)上推导出一个离散动力系统，并研究了其相关的转移算子p。我们的主要结果可以大致概括如下：我们显式构造了两个分段仿射函数u和v， Pu=u和Pv=β - 1v，使得对于每一个在[0,1]中支持且满足∫01Fdx=1的充分光滑F，我们有PkF=u+β - k(F(1)−F(0))v+o（β - k）在L∞上。这也与整数基的情况进行了比较，在整数基的情况下，更精细的渐近公式是可能的。

引用次数: 0

Nevai’s condition for measures with unbounded supports 具有无界支撑的测度的内瓦伊条件

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-09-03 DOI: 10.1016/j.jat.2025.106232

Grzegorz Świderski

We study Nevai’s condition from the theory of orthogonal polynomials on the real line. We prove that a large class of measures with unbounded Jacobi parameters satisfies Nevai’s condition locally uniformly on the support of the measure away from a finite explicit set. This allows us to give applications to relative uniform and weak asymptotics of Christoffel–Darboux kernels on the diagonal and to limit theorems for unconventionally normalized global linear statistics of orthogonal polynomial ensembles.

从实线上的正交多项式理论出发，研究了newai条件。证明了一类具有无界Jacobi参数的测度在远离有限显式集的测度支持上局部一致地满足Nevai条件。这允许我们给出对角线上Christoffel-Darboux核的相对一致和弱渐近的应用，以及正交多项式集合的非常规归一化全局线性统计的极限定理。

引用次数: 0

Asymptotics of the Humbert functions Ψ1 and Ψ2 Humbert函数的渐近性Ψ1和Ψ2

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-09-03 DOI: 10.1016/j.jat.2025.106233

Peng-Cheng Hang , Malte Henkel , Min-Jie Luo

A compilation of new results on the asymptotic behaviour of the Humbert functions

Ψ_{1}

and

Ψ_{2}

, and also on the Appell function

F_{2}

, is presented. As a by-product, we confirm a conjectured limit which appeared recently in the study of the

1 D

Glauber–Ising model. We also propose two elementary asymptotic methods and confirm through some illustrative examples that both methods have great potential and can be applied to a large class of problems of asymptotic analysis. Finally, some directions of future research are pointed out in order to suggest ideas for further study.

本文给出了关于Humbert函数Ψ1和Ψ2以及apell函数F2渐近行为的新结果汇编。作为一个副产品，我们证实了最近在一维格劳伯-伊辛模型研究中出现的一个推测极限。我们还提出了两种初等渐近方法，并通过一些例子证实了这两种方法都有很大的潜力，可以应用于大量的渐近分析问题。最后，对今后的研究方向进行了展望，为今后的研究提供思路。

引用次数: 0

Interpolation of compact multilinear operators between quasi-Banach spaces 拟巴拿赫空间间紧多线性算子的插值

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-08-22 DOI: 10.1016/j.jat.2025.106222

Fernando Cobos , Luz M. Fernández-Cabrera , Thomas Kühn

We investigate the interpolation properties of compact multilinear operators by the real method between quasi-Banach spaces. As an application we establish a reinforced version of a multilinear Marcinkiewicz theorem.

用实数方法研究了紧多线性算子在拟巴拿赫空间间的插值性质。作为一个应用，我们建立了一个增强版的多线性Marcinkiewicz定理。

引用次数: 0

Dedication 奉献

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-08-22 DOI: 10.1016/j.jat.2025.106231

Alexander Brudnyi, Natan Kruglyak, Mieczysław Mastyło, Paul Nevai, Amos Ron, Pavel Shvartsman

引用次数: 0

Invertible and Fredholm operators on interpolation scales 插值尺度上的可逆算子和Fredholm算子

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-08-20 DOI: 10.1016/j.jat.2025.106213

Irina Asekritova , Natan Kruglyak , Mieczysław Mastyło

We investigate the behaviour of invertible and Fredholm operators on interpolation scales constructed via a family of interpolation functors

{F_{θ}}_{θ \in (0, 1)}

. This family includes both complex and real interpolation functors. Our results demonstrate, in particular, that kernels and cokernels of operators are stable on intervals of parameters

θ

where the operators are Fredholm. Additionally, we introduce the notion of Fredholm operators in the category of Banach couples, establishing its relevance for the obtained results.

研究了由插值函子{Fθ}θ∈(0,1)构成的插值尺度上可逆算子和Fredholm算子的行为。这个族包括复插补函子和实插补函子。我们的结果特别证明了算子的核和核在算子为Fredholm的参数区间θ上是稳定的。此外，我们在Banach对的范畴中引入了Fredholm算子的概念，建立了它与所得结果的相关性。

引用次数: 0

A characterization of completely alternating functions 完全交替函数的表征

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-08-20 DOI: 10.1016/j.jat.2025.106230

Monojit Bhattacharjee , Rajkamal Nailwal

In this article, we characterize completely alternating functions on an abelian semigroup

S

in terms of completely monotone functions on the product semigroup

S \times Z_{+}

. We also discuss completely alternating sequences induced by a class of rational functions and obtain a set of sufficient conditions (in terms of its zeros and poles) to determine them. As an application, we show a complete characterization of several classes of completely monotone functions on

Z_{+}^{2}

induced by rational functions in two variables. We also derive a set of necessary conditions for the complete monotonicity of the sequence

{\prod_{i = 1}^{k} \frac{(n + a_{i})}{(n + b_{i})}}_{n \in Z_{+}}, a_{i}, b_{i} \in (0, \infty)

在本文中，我们用乘积半群S×Z+上的完全单调函数来表征阿贝尔半群S上的完全交替函数。我们还讨论了由一类有理函数诱导的完全交替序列，并得到了确定它们的一组充分条件（用它的零点和极点表示）。作为应用，我们给出了Z+2上由二元有理函数诱导的几类完全单调函数的完备刻划。我们还导出了序列{∏i=1k(n+ai)(n+bi)}n∈Z+,ai,bi∈(0,∞)的完全单调性的一组必要条件。

引用次数: 0

Runge-type approximation theorem for Banach-valued H∞ functions on a polydisk 多盘上banach值H∞函数的龙格逼近定理

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-08-19 DOI: 10.1016/j.jat.2025.106221

Alexander Brudnyi

<div><div>Let <math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>⊂</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math> denote the open unit polydisk, and let <math><mrow><mi>K</mi><mo>⊂</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math> be a Cartesian product of planar compacta. Let <math><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>̂</mo></mrow></mover><mo>⊂</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math> be an open neighborhood of the closure <math><mover><mrow><mi>K</mi></mrow><mrow><mo>̄</mo></mrow></mover></math> in <math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math>, where <math><mi>M</mi></math> is the maximal ideal space of the algebra <math><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup></math> of bounded holomorphic functions on <math><mi>D</mi></math>. Given a complex Banach space <math><mi>X</mi></math>, denote by <math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></math> the Banach space of bounded <math><mi>X</mi></math>-valued holomorphic functions on an open set <math><mrow><mi>V</mi><mo>⊂</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math>.</div><div>We prove that any <math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></math>, where <math><mrow><mi>U</mi><mo>=</mo><mover><mrow><mi>U</mi></mrow><mrow><mo>̂</mo></mrow></mover><mo>∩</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math>, can be uniformly approximated on <math><mi>K</mi></math> by functions of the form <math><mrow><mi>h</mi><mo>/</mo><mi>b</mi></mrow></math>, where <math><mrow><mi>h</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></math> and <math><mi>b</mi></math> is a finite product of interpolating Blaschke products satisfying <math><mrow><msub><mrow><mo>inf</mo></mrow><mrow><mi>K</mi></mrow></msub><mrow><mo>|</mo><mi>b</mi><mo>|</mo></mrow><mo>></mo><mn>0</mn></mrow></math>. Moreover, if <math><mover><mrow><mi>K</mi></mrow><mrow><mo>̄</mo></mrow></mover></math> is contained in a compact holomorphically convex subset of <math><mover><mrow><mi>U</mi></mrow><mrow><mo>̂</mo></mrow></mover></math>, then such approximations can be achieved without denominators: that is, <math><mi>f</mi></math> can be approximated uniformly on <math><mi>K</mi></math>

设Dn∧∧n表示开单位多盘，设K∧n是平面紧化的笛卡尔积。设Û∧Mn是Mn中闭包K的一个开邻域，其中M是d上有界全纯函数的代数H∞的最大理想空间。给定一个复巴拿赫空间X，用H∞（V，X）表示开集V上有界X值全纯函数的巴拿赫空间。证明了任意f∈H∞(U，X)，其中U=Û∩Dn，可以用H /b形式的函数在K上一致逼近，其中H∈H∞（Dn，X）与b是满足infK|b|>；0的插值Blaschke积的有限积。此外，如果K∈包含在Û的紧全纯凸子集中，则这种近似可以不带分母地实现：即f可以由H∞（Dn，X）的元素在K上一致地近似。这些结果，本质上是由本文的主要贡献：单位盘d的开子集上的banach值全纯函数的一个新的建设性runge型逼近定理所得到的。我们的工作扩展了Suárez关于M的紧子集上解析芽的逼近的基本结果，并为经典的关于对于所有n≥2的H∞(Dn)的最大理想空间中Dn是否密集的问题提供了新的视角。

引用次数: 0

Widom factors in ℂn 智慧因子

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-08-19 DOI: 10.1016/j.jat.2025.106227

Gökalp Alpan , Turgay Bayraktar , Norm Levenberg

We generalize the theory of Widom factors to the

ℂ^{n}

setting. We define Widom factors of compact subsets

K \subset ℂ^{n}

associated with multivariate orthogonal polynomials and weighted Chebyshev polynomials. We show that on product subsets

K = K_{1} \times \dots \times K_{n}

ℂ^{n}

, where each

K_{j}

is a non-polar compact subset of

ℂ

, these quantities have universal lower bounds which directly extend one dimensional results. Under the additional assumption that each

K_{j}

is a subset of the real line, we provide improved lower bounds for Widom factors for some weight functions

w

; in particular, for the case

w \equiv 1

. Finally, we define the Mahler measure of a multivariate polynomial relative to

K \subset ℂ^{n}

and obtain lower bounds for this quantity on product sets.

我们将智慧因子的理论推广到基于n的集合。我们定义了与多元正交多项式和加权切比雪夫多项式相关的紧子集K∧n的智能因子。我们证明了在n的乘积子集K= k1x⋯×Kn上，其中每个Kj是的非极紧子集，这些量具有直接扩展一维结果的普遍下界。在附加的假设下，每个Kj是实线的一个子集，我们为一些权重函数w提供了改进的智能因子下界；特别地，对于w≡1。最后，我们定义了一个多元多项式相对于K∧n的马勒测度，并得到了这个量在积集上的下界。

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引用次数: 0

On the extremal eigenvalues of Jacobi ensembles at zero temperature 零度下雅可比系综的极值特征值

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-08-19 DOI: 10.1016/j.jat.2025.106229

Kilian Hermann, Michael Voit

For the

β

-Hermite, Laguerre, and Jacobi ensembles of dimension

N

there exist central limit theorems for the freezing case

β \to \infty

such that the associated means and covariances can be expressed in terms of the associated Hermite, Laguerre, and Jacobi polynomials of order

N

respectively as well as via the associated dual polynomials in the sense of de Boor and Saff. In this paper we derive limits for

N \to \infty

for the covariances of the

r \in N

largest (and smallest) eigenvalues for these frozen Jacobi ensembles in terms of Bessel functions. These results correspond to the hard edge analysis in the frozen Laguerre cases by Andraus and Lerner-Brecher and to known results for finite

β

对于N维的β-Hermite、Laguerre和Jacobi系综，存在冻结情况下β→∞的中心极限定理，使得相关均值和协方差可以分别用相关的N阶Hermite、Laguerre和Jacobi多项式表示，也可以通过de Boor和Saff意义上的相关对偶多项式表示。本文用贝塞尔函数导出了这些冻结Jacobi系综的r∈N的最大（和最小）特征值的协方差在N→∞时的极限。这些结果与Andraus和Lerner-Brecher在冰冻Laguerre案例中的硬边分析以及有限β的已知结果相对应。

引用次数: 0

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