Cesaro Summability and Marchaud Inequality

For Fourier expansions it will be shown that sharp Marchaud-type inequalities follow from the positivity of the Cesaro summability of some order. Some results of sharp Marchaud-type will be derived using this method when other known methods cannot be used.     

Characterization of a Two-Weighted Vector-Valued Inequality for Fractional Maximal Operators

We give a characterization of the weights u(·) and v(·) for which the fractional maximal operator M _s is bounded from the weighted Lebesgue spaces L ^p(l ^r, vdx) into L ^q(l ^r, udx) whenever 0 s < n, 1 < p, r < , and 1 q < .     

约束图像拓扑下的向量值拟变分不等式解集的通有稳定性

主要运用研究通有性质的方法研究向量值拟变分不等式解的稳定性.首先引入约束映射在图像拓扑意义下的Hausdorff度量,这是一种有别于通常一致度量的新度量,然后在此弱图像拓扑下,给出并证明了关于向量值拟变分不等式解的通有稳定性的几个结论.结论表明,在Baire分类的意义下,大多数的向量值拟变分不等式问题的解关于新定义的度...     

Vector-Valued Cosine Transforms

The aim of this paper is to characterize cosine and sine functions in terms of vector-valued cosine transforms on a Banach space X. We also introduce vector-valued sine transforms.     

Summability in of a vector measure

We show a picture of the relations among different types of summability of series in the space  of integrable functions with respect to a vector measure m of relatively norm compact range. In order to do that, we study the class of the so‐called m‐1‐summing operators. We give several applications regarding the existence of copies of c ₀ in , as well as on m‐1‐summing operators which are weakly compact, Asplund or weakly precompact.     

Summability and vector amarts

We show that an operator is absolutely summing if and only if it maps amarts into uniform amarts, from which we can deduce a theorem of A. Bellow and another of Edgar-Sucheston. We also show that the absolute value of a Banach lattice valued potential is a potential if and only if the lattice is an A-M space from which we deduce that the L¹-bounded amarts form a Ricsz space if and only if the space is finite dimensional.     

Vector-Valued capacities

It is well-known that ifX is a compact metric space and ifI is a capacity onX then every analytic subset ofX isI-capacitable [2], [1]. We introduce the notion of vector-valued capacity in the case when the codomain is a Banach lattice and we prove an analogous theorem for analytic subsets of a Polish space. Finally, we show that every vectorvalued outer measure is a capacity and, in connection with the so-called “marginal problem”, we give an example of a capacity taking values in a reflexive Banach lattice.     

Frame Multiplication Theory and a Vector-Valued DFT and Ambiguity Function

Vector-valued discrete Fourier transforms (DFTs) and ambiguity functions are defined. The motivation for the definitions is to provide realistic modeling of multi-sensor environments in which a useful time–frequency analysis is essential. The definition of the DFT requires associated uncertainty principle inequalities. The definition of the ambiguity function requires a component that leads to formulating a mathematical theory in which two essential algebraic operations can be made compatible in a natural way. The theory is referred to as frame multiplication theory. These definitions, inequalities, and theory are interdependent, and they are the content of the paper with the centerpiece being frame multiplication theory. The technology underlying frame multiplication theory is the theory of frames, short time Fourier transforms, and the representation theory of finite groups. The main results have the following form: frame multiplication exists if and only if the finite frames that arise in the theory are of a certain type, e.g., harmonic frames, or, more generally, group frames. In light of the complexities and the importance of the modeling of time-varying and dynamical systems in the context of effectively analyzing vector-valued multi-sensor environments, the theory of vector-valued DFTs and ambiguity functions must not only be mathematically meaningful, but it must have constructive implementable algorithms, and be computationally viable. This paper presents our vision for resolving these issues, in terms of a significant mathematical theory, and based on the goal of formulating and developing a useful vector-valued theory.

     

On a New Summability Result for Operator Matrices

For matrices of quasi-homogeneous operators we establish a summability result which is a useful complement of a recent summability result.     

Summability of Hermite Series

Let f∈L ^p (R), 1≤p≤t8, and c _j be the inner product of f and the Hermite function h _j . Assume that c _j 's satisfy $\left| {c_j } \right| \cdot f = o\left( 1 \right)\;\quad as\;j \to \infty $ If r=5/4, then the Hermite series Σc _j h _j conerges to f almost everywhere. If r=9/4-1/p, the Σ c _j h _j converges to f in L ^p (R).     

Summability in topological spaces

The main purpose of the paper is to introduce the notion of summability in abstract Hausdorff topological spaces. We give a characterization of such summability methods when the space allows a countable base. We also provide several Tauberian theorems in topological structures. Some open problems are discussed.     

Summability of Hermite Series

Let fL ^p (R), 1pt8, and c _j be the inner product of f and the Hermite function h _j . Assume that c _j 's satisfy If r=5/4, then the Hermite series c _j h _j conerges to f almost everywhere. If r=9/4-1/p, the c _j h _j converges to f in L ^p (R).     

向量值哈代-洛伦茨空间

对于1r∞与巴拿哈空间B=L~r(Ω,F,μ),我们研究了欧几里得空间R~n上B-值缓分布构成的哈代-洛伦茨空间H~(p,q)(R~n,B)及哈代-洛伦茨空间之间的内插,其中0p∞和0q≤∞,获得了H~(p,q)(R~n,B)的一系列等价的刻画及其原子分解.若Ω={1},则H~(p,q)(R~n,B)=H~(p,q)(R~n)是经典的情形;若Ω=Z是整数集且μ是Z上的计数测度并且r=2,0p∞及q=∞,则H~(p,q)(R~n,B)=H~(p,∞)(R~n,e~2)转化为Grafakos和He在文[Weak Hardy spaces,Preprint,2014]中讨论的情形.     

Matrix Summability and Positive Linear Operators

In the present paper we prove a Korovkin type approximation theorem for a sequence of positive linear operators acting from a weighted space C_ρ1 into a weighted space B_ρ2 with the use of a matrix summability method which includes both convergence and almost convergence. We also study the rates of convergence of these operators.     

Summability of Laguerre expansions

В РАБОтЕ пОлУЧЕНы УсИ лЕНИь РЕжУльтАтОВ МА РкЕттА О сУММИРУЕМОстИ МОДИФ ИцИРОВАННых РАжлОжЕНИИ лАгЕРРА. У стАНОВлЕНО, ЧтОα=1/6 Ест ь кРИтИЧЕскИИ ИНДЕкс Д ль сУММИРУЕМОстИ пО ЧЕжАРО. ДОкАжАНО, Чт О пРИα=1/6 сРЕДНИЕ ЧЕжАР О схОДьтсь пОЧтИ ВсУДУ. пОлУЧЕН тАкжЕ АНАлОг тЕОРЕМы ФЕИЕРА-лЕБЕгА.