From weak to strong types of $L_E^1$-convergence by the Bocce-criterion

Criterion for convergence almost everywhere, with applications

We derive the necessary and sufficient condition for almost sure convergence of the sequence of measurable functions, and consider some applications in the theory of Fourier series and in the theory of random fields.

A new metastable convergence criterion and an application in the theory of uniformly convex Banach spaces

We study a convergence criterion which generalises the notion of being monotonically decreasing, and introduce a quantitative version of this criterion, a so called metastable rate of asymptotic decreasingness. We then present a concrete application in the fixed point theory of uniformly convex Banach spaces, in which we carry out a quantitative analysis of a convergence proof of Kirk and Sims. More precisely, we produce a rate of metastability (in the sense of Tao) for the Picard iterates of mappings T which satisfy a variant of the convergence criterion, and whose fixed point set has nonempt…

椭圆变分不等式的一个收敛准则

A Convergence Criterion for Elliptic Variational Inequalities

本文针对Hilbert空间中带单侧约束的椭圆变分不等式，提出了保证序列收敛于唯一解的充分必要条件，并应用于恢复适定性结果及解决热传导与弹性无摩擦接触问题。

弱贪婪算法的一个闭式收敛准则

A closed-form convergence criterion for the weak greedy algorithm

针对希尔伯特空间中的弱贪婪算法，提出一个等价的闭式收敛准则：算法收敛当且仅当无穷级数∑(1+n∑t_k^2)^{-1/2}t_n^2发散。

Sobolev regularity and an enhanced Jensen inequality

We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}(\R^n)$. This criterion consists of comparing the value of a functional $\int f(u)$ with the values of the same functional applied to convolutions of $u$ with a Dirac sequence. The difference of these values converges to zero as the convolutions approach $u$, and we prove that the rate of convergence to zero is connected to regularity: $u\in W^{1,2}$ if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization problem with constraints, where regularity of minimiz…