Strong uniform continuity

Let be an ideal of subsets of a metric space X,d. This paper considers a strengthening of the notion of uniform continuity of a function restricted to members of which reduces to ordinary continuity when consists of the finite subsets of X and agrees with uniform continuity on members of when is either the power set of X or the family of compact subsets of X. The paper also presents new function space topologies that are well suited to this strengthening. As a consequence of the general theory, we display necessary and sufficient conditions for continuity of the pointwise limit of a net of continuous functions.     

Almost uniform convergence

The almost uniform convergence is between uniform and quasi-uniform one. We give some necessary and sufficient conditions under which the almost uniform convergence coincides on compact sets with uniform, quasi-uniform or uniform convergence, respectively. In the second section continuity of almost uniform limits is considered. Finally we characterize the set of all points at which a net of functions is almost uniformly convergent to a given function.     

函数列一致收敛判别法

利用函数列和函数一致连续的有关性质,得到了函数列一致收敛新的判别法.     

About the statistical uniform convergence

In this work we study the concept of statistical uniform convergence. We generalize some results of uniform convergence in double sequences to the case of statistical convergence. We also prove a basic matrix theorem with statistical convergence.     

Persistent convergence on randomly deleted sets

At time t_k, a unit with magnitude X_k and lifetime L_k enters a system. Let λ be a real valued function on the finite real sequences. One such sequence, B*_t, consists of the X_k's for which t_k t < t_k + L_k. When λ(X₁,…, X_n) converges (in some sense) to φ, we find conditions under which λ(B*_t) converges or fails to converge to φ in the same sense.     

On the strong convergence of derivatives in a time optimal problem

We consider a time optimal problem for a system described by a differential inclusion, whose right hand side is not necessarily convex valued. Under the assumption of strict convexity of the map obtained by convexifying the original, non-convex valued map, we obtain the strong convergence of the derivatives of any uniformly converging minimizing sequence. The assumptions required by this result are satisfied, for instance, by the classical brachystochrone problem and by Fermat’s principle.     

A criterion for the uniform convergence of sinc-approximations on a segment

In this paper we obtain a criterion for the uniform convergence inside the interval (0, π) of values of E. T. Whittaker operators

$L_n (f,x) = \sum\limits_{k = 0}^n {\frac{{sin(nx - k\pi )}}{{nx - k\pi }}f\left( {\frac{{k\pi }}{n}} \right)} $

for continuous functions. This criterion is similar to that of A. A. Privalov for the convergence of interpolation Lagrange-Chebyshev polynomials and trigonometric ones.

     

On the strong convergence rate for positively associated random variables

For sequences of the positively associated random variables which are strictly weaker than the classical associated random ones introduced by Esary et al. (1967) [7], strong convergence rate is obtained, which reaches the available one for independent random variables in terms of Berstein type inequality. Further, we give the corresponding precise asymptotics with respect to the rate mentioned above, which extend and improve the relevant results in Fu (2009) [8].     

A general law of moment convergence rates for uniform empirical process

Let {X _n ; n ≥ 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process $F_n (t) = n^{ - \tfrac{1} {2}} \sum\nolimits_{i = 1}^n {(I_{\{ X_i \leqslant t\} } - t),0} \leqslant t \leqslant 1,\left\| {F_n } \right\| = \sup _{0 \leqslant t \leqslant 1} \left| {F_n (t)} \right| $F_n (t) = n^{ - \tfrac{1} {2}} \sum\nolimits_{i = 1}^n {(I_{\{ X_i \leqslant t\} } - t),0} \leqslant t \leqslant 1,\left\| {F_n } \right\| = \sup _{0 \leqslant t \leqslant 1} \left| {F_n (t)} \right| . In this paper, the exact convergence rates of a general law of weighted infinite series of E {‖F _n ‖ − ɛg ^s (n)}₊ are obtained.     

On necessary and sufficient conditions for uniform strong consistency of estimators of a density and its derivatives

Based on a random sample from a population with (unknown) probability density f, this note exhibits a class of statistics f^(p) for each fixed integer $\text{p ≧ 0}$. It is shown that f^(p) are uniformly strongly consistent estimators of f^(p), the pth order derivative of f, if and only iff^(p)is bounded and uniformly continuous.     

A strong convergence theorem for solutions to a nonhomogeneous second order evolution equation

In this paper, we establish the strong convergence of possible solutions to the following nonhomogeneous second order evolution system

     

Dini定理修正

讨论经典Dini定理,证明当Dini定理的连续性条件被放宽成"un(x)一u(x)有最大值"时,同样可以得到一致收敛的结论.在函数项级数和含参变量积分中,Dini定理可以进行类似的修改.最后,给出修正后的Dini定理的一个具体应用.     

A note on the rate of convergence in the strong law of large numbers for martingales

We prove a Baum-Katz-Nagaev type rate of convergence in the Marcinkiewicz-Zygmund and Kolmogorov strong laws of large numbers for norm bounded martingale difference sequences.     

Local convergence of some iterative methods for generalized equations

We study generalized equations of the following form:

(render)

0f(x)+g(x)+F(x),

where f is Fréchet differentiable in a neighborhood of a solution x^* of (*) and g is Fréchet differentiable at x^* and where F is a set-valued map acting in Banach spaces. We prove the existence of a sequence (x_k) satisfying

which is super-linearly convergent to a solution of (*). We also present other versions of this iterative procedure that have superlinear and quadratic convergence, respectively.     

Boolean algebras and uniform convergence of series

Several classical results on uniform convergence of unconditionally Cauchy series are generalized to weakly unconditionally Cauchy series. This uniform convergence is characterized through subalgebras and subfamilies of . A generalization of the Orlicz-Pettis theorem is also proved by mean of subalgebras of .     

Absolute and uniform convergence of eigenfunction expansions of integral operators with kernels admitting derivative discontinuities on the diagonals

An analog of Szasz’s theorem on the absolute convergence of trigonometric Fourier series is established for expansions in the eigen and associated functions of integral operators some of whose kernels involve derivatives with discontinuities on the diagonals.