有限开覆盖定理通过化“无限和运算”为“有限和运算”的方式,将一个局部成立的性质推广到在一个闭区间上整体成立。本文从整体与局部的关系分析了有限开覆盖定理的内涵,并给出具体例子分析此定理在实现整体性质与局部性质相互转化方面...有限开覆盖定理通过化“无限和运算”为“有限和运算”的方式,将一个局部成立的性质推广到在一个闭区间上整体成立。本文从整体与局部的关系分析了有限开覆盖定理的内涵,并给出具体例子分析此定理在实现整体性质与局部性质相互转化方面以及在实际问题中的应用。The finite open cover theorem extends a property that holds locally to the entire closed interval by transforming “infinite sum operations” into “finite sum operations.” This paper analyzes the essence of the finite open cover theorem from the perspective of the relationship between the global and the local. It provides concrete examples to illustrate how the theorem facilitates the mutual transformation of global and local properties, as well as its applications in practical problems.展开更多
This paper explores the recovery of block sparse signals in frame-based settings using the l_(2)/l_(q)-synthesis technique(0<q≤1).We propose a new null space property,referred to as block D-NSP_(q),which is based ...This paper explores the recovery of block sparse signals in frame-based settings using the l_(2)/l_(q)-synthesis technique(0<q≤1).We propose a new null space property,referred to as block D-NSP_(q),which is based on the dictionary D.We establish that matrices adhering to the block D-NSP_(q)condition are both necessary and sufficient for the exact recovery of block sparse signals via l_(2)/l_(q)-synthesis.Additionally,this condition is essential for the stable recovery of signals that are block-compressible with respect to D.This D-NSP_(q)property is identified as the first complete condition for successful signal recovery using l_(2)/l_(q)-synthesis.Furthermore,we assess the theoretical efficacy of the l2/lq-synthesis method under conditions of measurement noise.展开更多
Inhomogeneous Calderon-Zygmund operator T maps each atom into an approximate molecule of weighted local Hardy space if and only if some approximate cancellation condition holds for T.An equivalent norm for weighted Le...Inhomogeneous Calderon-Zygmund operator T maps each atom into an approximate molecule of weighted local Hardy space if and only if some approximate cancellation condition holds for T.An equivalent norm for weighted Lebesgue space which has vanishing moments up to order s plays an important role,where s∈N.展开更多
设μρ,D,{ nk}是由以下离散测度的无限卷积定义的Borel概率测度:μρ,D,{ nk}=δρn1D∗δρn2D∗δρn3D∗⋯,其中0ρ1,D⊂ℝ是一有限集,{ nk}k=1∞是一个严格递增的正整数序列,且supk≥1{ nk+1−nk}∞。本文证明了:若Z(δ^D)包含在Lattice...设μρ,D,{ nk}是由以下离散测度的无限卷积定义的Borel概率测度:μρ,D,{ nk}=δρn1D∗δρn2D∗δρn3D∗⋯,其中0ρ1,D⊂ℝ是一有限集,{ nk}k=1∞是一个严格递增的正整数序列,且supk≥1{ nk+1−nk}∞。本文证明了:若Z(δ^D)包含在Lattice集中且对任意的r>1,无理数ρ∈ℚ1/r:={ ρ=u1/r:0u1 是一有理数 },那么μρ,D,{ nk}不是谱测度。Let μρ,D,{ nk}be a Borel probability measure defined by the following infinite convolution of discrete measures: μρ,D,{ nk}=δρn1D∗δρn2D∗δρn3D∗⋯,where 0ρ1, D⊂ℝis a finite set, and{ nk}k=1∞is a strictly increasing sequence of positive integers with supk≥1{ nk+1−nk}∞. In this paper, we will show that if Z(δ^D)is contained in a Lattice set and for any r>1, the irrational number ρ∈ℚ1/r:={ ρ=u1/r:0u1 is rational }, then μρ,D,{ nk}is not a spectral measure.展开更多
周宏毅在论文关于Herman环与临界点中给出了三次有理函数且其临界点严格位于Herman环的边界分支的例子。该构造中主要用到临界点都位于单位圆周且保持单位圆周不动的有理函数的存在性。本文给出了一般的有理函数临界点均在单位圆周且保...周宏毅在论文关于Herman环与临界点中给出了三次有理函数且其临界点严格位于Herman环的边界分支的例子。该构造中主要用到临界点都位于单位圆周且保持单位圆周不动的有理函数的存在性。本文给出了一般的有理函数临界点均在单位圆周且保持单位圆周不动的存在性证明。同时讨论了一般显示构造的方法。In A Note on Herman, Hongyi Zhou gave an example of a cubic rational function whose critical points strictly lie on the boundary of the Herman ring. The construction mainly relies on the existence of rational functions whose critical points are located on the unit circle and keep the unit circle invariant. In this paper, we provide a general proof for the existence of rational functions whose critical points are all on the unit circle and keep the unit circle invariant. Additionally, we discuss the general methods for explicit constructions.展开更多
文摘有限开覆盖定理通过化“无限和运算”为“有限和运算”的方式,将一个局部成立的性质推广到在一个闭区间上整体成立。本文从整体与局部的关系分析了有限开覆盖定理的内涵,并给出具体例子分析此定理在实现整体性质与局部性质相互转化方面以及在实际问题中的应用。The finite open cover theorem extends a property that holds locally to the entire closed interval by transforming “infinite sum operations” into “finite sum operations.” This paper analyzes the essence of the finite open cover theorem from the perspective of the relationship between the global and the local. It provides concrete examples to illustrate how the theorem facilitates the mutual transformation of global and local properties, as well as its applications in practical problems.
基金Supported by The Featured Innovation Projects of the General University of Guangdong Province(2023KTSCX096)The Special Projects in Key Areas of Guangdong Province(ZDZX1088)Research Team Project of Guangdong University of Education(2024KYCXTD018)。
文摘This paper explores the recovery of block sparse signals in frame-based settings using the l_(2)/l_(q)-synthesis technique(0<q≤1).We propose a new null space property,referred to as block D-NSP_(q),which is based on the dictionary D.We establish that matrices adhering to the block D-NSP_(q)condition are both necessary and sufficient for the exact recovery of block sparse signals via l_(2)/l_(q)-synthesis.Additionally,this condition is essential for the stable recovery of signals that are block-compressible with respect to D.This D-NSP_(q)property is identified as the first complete condition for successful signal recovery using l_(2)/l_(q)-synthesis.Furthermore,we assess the theoretical efficacy of the l2/lq-synthesis method under conditions of measurement noise.
文摘Inhomogeneous Calderon-Zygmund operator T maps each atom into an approximate molecule of weighted local Hardy space if and only if some approximate cancellation condition holds for T.An equivalent norm for weighted Lebesgue space which has vanishing moments up to order s plays an important role,where s∈N.
文摘设μρ,D,{ nk}是由以下离散测度的无限卷积定义的Borel概率测度:μρ,D,{ nk}=δρn1D∗δρn2D∗δρn3D∗⋯,其中0ρ1,D⊂ℝ是一有限集,{ nk}k=1∞是一个严格递增的正整数序列,且supk≥1{ nk+1−nk}∞。本文证明了:若Z(δ^D)包含在Lattice集中且对任意的r>1,无理数ρ∈ℚ1/r:={ ρ=u1/r:0u1 是一有理数 },那么μρ,D,{ nk}不是谱测度。Let μρ,D,{ nk}be a Borel probability measure defined by the following infinite convolution of discrete measures: μρ,D,{ nk}=δρn1D∗δρn2D∗δρn3D∗⋯,where 0ρ1, D⊂ℝis a finite set, and{ nk}k=1∞is a strictly increasing sequence of positive integers with supk≥1{ nk+1−nk}∞. In this paper, we will show that if Z(δ^D)is contained in a Lattice set and for any r>1, the irrational number ρ∈ℚ1/r:={ ρ=u1/r:0u1 is rational }, then μρ,D,{ nk}is not a spectral measure.
文摘周宏毅在论文关于Herman环与临界点中给出了三次有理函数且其临界点严格位于Herman环的边界分支的例子。该构造中主要用到临界点都位于单位圆周且保持单位圆周不动的有理函数的存在性。本文给出了一般的有理函数临界点均在单位圆周且保持单位圆周不动的存在性证明。同时讨论了一般显示构造的方法。In A Note on Herman, Hongyi Zhou gave an example of a cubic rational function whose critical points strictly lie on the boundary of the Herman ring. The construction mainly relies on the existence of rational functions whose critical points are located on the unit circle and keep the unit circle invariant. In this paper, we provide a general proof for the existence of rational functions whose critical points are all on the unit circle and keep the unit circle invariant. Additionally, we discuss the general methods for explicit constructions.
