Tauberian conditions for w-almost convergent double sequences

In this paper, we introduce the concept of w-almost convergent sequences. Such a definition is a weak form of almost convergent sequences given by G. G. Lorentz in [Acta Math. 80(1948),167-190]. We give a detailed study on w-almost convergent double sequences and prove that w-almost convergence and almost convergence are equivalent under the boundedness of the given sequence. The Tauberian results for w-almost convergence are established. Our Tauberian results generalize a result of Lorentz and Tauber’s second theorem. Moreover, we prove that w-almost convergence and norm convergence are equivalent for the sequence of the rectangular partial sums of the Fourier series of f ∈ L^p(T²), where 1 < p < ∞.      

On the invariant mean and statistical convergence

Two concepts - one of almost convergence and the other of statistical convergence - play a very active role in recent research on summability theory. The definition of almost convergence introduced by Lorentz [G.G. Lorentz, A contribution to theory of divergent sequences, Acta Math. 80 (1948) 167–190] originated from the concept of the Banach limit, while the statistical convergence introduced by Fast [H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244] was defined through the concept of density. Both involve non-matrix methods of summability and they are incompatible. In this work we define two new kinds of summability methods by using these two mutually incompatible concepts of the Banach limit and of density to deal with those sequences which are statistically convergent but not almost convergent or vice versa.     

An application of almost convergence in approximation theorems

The concept of almost convergence was introduced by Lorentz in 1948 [G.G. Lorentz, A contribution to theory of divergent sequences, Acta Math. 80 (1948) 167–190], and has various applications. In this work we apply this method to prove some Korovkin type approximation theorems.     

Weighted Hardy inequalities for decreasing sequences and functions

We obtain a complete characterization of the weights for which Hardy's inequality holds on the cone of non-increasing sequences. Our proofs translate immediately to the analogous inequality for non-increasing functions, thereby also completing the investigation in that direction. As an application of our results we characterize the boundedness of the Hardy-Littlewood maximal operator on Lorentz sequence spaces.     

Tauberian conditions for almost convergence

In [Acta Math. 80(1948), 167–190], G. G. Lorentz characterized almost convergent sequences in (or in ) in terms of the concept of uniform convergence of the de la Vallée-Poussin means. In this paper, we present Tauberian results which relate almost convergence to norm convergence or to the (C, 1) convergence. Our results generalize Kronecker lemma. As a consequence, we prove that almost convergence and norm convergence are equivalent for the sequence of the partial sums of the Fourier series of (or ), where . We also show that our results can be used to derive Fatou’s theorem.      

On vector-valued Banach limits

In this brief communication we propose a vector-valued version of Lorentz’ intrinsic characterization of almost convergence, for which we find a legitimate extension of the concept of Banach limit to vector-valued sequences. Banach spaces 1-complemented in their biduals admit vector-valued Banach limits, whereas c ₀ does not.     

A note on almost sure uniform and complete convergences of a sequence of random variables

The aim of this note is to introduce another way of defining the almost sure uniform convergence, which is necessary when studying some mathematical results on the existence of price bubbles in certain scenarios of trading securities. This mode of convergence of random variables' sequences is intermediate between the uniform and the almost sure ones, and, more specifically, between the uniform and the complete convergences. In this way, this paper presents some mathematical characterizations of both almost sure uniform and complete convergences, and shows that the almost sure uniform convergence is a particular case of complete convergence, when the number of summands in the series defining this mode of convergence is finite. Finally, this paper presents the relation of almost surely uniform convergence with convergence in mean when the random variable limit is integrable. Moreover, almost surely convergence and local boundedness of the sequence of random variables minus its limit are sufficient to derive convergence in mean.     

Convolution Inequalities in Lorentz Spaces

In this paper we study boundedness of the convolution operator in different Lorentz spaces. We obtain the limit case of the Young-O’Neil inequality in the classical Lorentz spaces. We also investigate the convolution operator in the weighted Lorentz spaces.     

Almost-convergent sequences with respect to polynomial hypergroups

Each polynomial hypergroup on ?₀ generates a family of generalized translation operators?T _m on sequence spaces. We introduce the concept of almost convergence for polynomial hypergroups (determined by the operators?T _m ), extending the notion of almost convergence introduced by Lorentz. Our investigations lead to two theorems characterizing almost convergent sequences on polynomial hypergroups.     

Global convergence of an SQP method without boundedness assumptions on any of the iterative sequences

Usual global convergence results for sequential quadratic programming (SQP) algorithms with linesearch rely on some a priori assumptions about the generated sequences, such as boundedness of the primal sequence and/or of the dual sequence and/or of the sequence of values of a penalty function used in the linesearch procedure. Different convergence statements use different combinations of assumptions, but they all assume boundedness of at least one of the sequences mentioned above. In the given context boundedness assumptions are particularly undesirable, because even for non-pathological and well-behaved problems the associated penalty functions (whose descent is used to produce primal iterates) may not be bounded below for any value of the penalty parameter. Consequently, boundedness assumptions on the iterates are not easily justifiable. By introducing a very simple and computationally cheap safeguard in the linesearch procedure, we prove boundedness of the primal sequence in the case when the feasible set is nonempty, convex, and bounded. If, in addition, the Slater condition holds, we obtain a complete global convergence result without any a priori assumptions on the iterative sequences. The safeguard consists of not accepting a further increase of constraints violation at iterates which are infeasible beyond a chosen threshold, which can always be ensured by the proposed modified SQP linesearch criterion. The author is supported in part by CNPq Grants 301508/2005-4, 490200/2005-2, 550317/2005-8, by PRONEX–Optimization, and by FAPERJ Grant E-26/151.942/2004.     

On some generalizations of Lorentz's almost convergence

We investigate an extension of the almost convergence of G.G. Lorentz, further weakening the notion of M-almost convergence we defined in [S. Mercourakis, G. Vassiliadis, An extension of Lorentz's almost convergence and applications in Banach spaces, Serdica Math. J. 32 (2006) 71–98] and requiring that the means of a bounded sequence restricted on a subset M of converge weakly in ℓ^∞(M). The case when M has density 1 is of special interest and in this case we derive a result in the direction of the Mean Ergodic Theorem (see Theorem 2).     

级联算法在Besov和Triebel-Lizorkin空间上的有界性和收敛性(Ⅱ)(英文)

本文利用联合谱半径刻画了级联算法在Ｂｅｓｏｖ和Ｔｈｉｅｂｅｌ－Ｌｉｚｏｒｋｉｎ空间上的收敛性,给出了级联算法初值函数矩条件的新证明,并利用到细分分布的光滑性和非齐次细分方程解的存在性等方面．特别地,在某些条件下,我们证明了级联算法的有界性和收敛性相互等价．     

Sharp Lorentz Estimates for Dyadic-like Maximal Operators and Related Bellman Functions

We precisely evaluate Bellman-type functions for the dyadic maximal operator on \(\mathbb {R}^{n}\) and of maximal operators on martingales related to local Lorentz-type estimates. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors, we precisely evaluate the supremum of the Lorentz quasinorm of the maximal operator on a function \(\phi \) when the integral of \(\phi \) is fixed and also the same Lorentz quasinorm of \(\phi \) is fixed. Also we find the corresponding supremum when the integral of \(\phi \) is fixed and several weak type conditions are given.     

分片线性谱序列生成的广义Fourier级数的Lp收敛性

In the present paper, we discuss some properties of piecewise linear spectral sequences introduced by Liu and Xu. We have a study on the pointwise and almost everywhere convergence of its corresponding series. Also, it is shown that the set (G) constructed from piecewise linear spectral sequences are bases, but not unconditional bases, for LP(0,1) where 1 ＜ p ＜∞, p ≠ 2.     

Maximal and Riesz Potential Operators in Double Phase Lorentz Spaces of Variable Exponents

Mathematical Notes - In the present note, we discuss the boundedness of maximal and Riesz potential operators in double-phase Lorentz spaces of variable exponents defined by a symmetric decreasing...     

Almost lacunary statistical and strongly almost lacunary convergence of sequences of fuzzy numbers

The purpose of this paper is to introduce the concepts of almost lacunary statistical convergence and strongly almost lacunary convergence of sequences of fuzzy numbers. We give some relations related to these concepts. We establish some connections between strongly almost lacunary convergence and almost lacunary statistical convergence of sequences of fuzzy numbers. It is also shown that if a sequence of fuzzy numbers is strongly almost lacunary convergent with respect to an Orlicz function then it is almost lacunary statistical convergent.     

Global Convergence of the Newton Interior-Point Method for Nonlinear Programming

The aim of this paper is to show that the theorem on the global convergence of the Newton interior–point (IP) method presented in Ref. 1 can be proved under weaker assumptions. Indeed, we assume the boundedness of the sequences of multipliers related to nontrivial constraints, instead of the hypothesis that the gradients of the inequality constraints corresponding to slack variables not bounded away from zero are linearly independent. By numerical examples, we show that, in the implementation of the Newton IP method, loss of boundedness in the iteration sequence of the multipliers detects when the algorithm does not converge from the chosen starting point.     

CONVERGENCE OF ONLINE GRADIENT METHOD WITH A PENALTY TERM FOR FEEDFORWARD NEURAL NETWORKS WITH STOCHASTIC INPUTS

Online gradient algorithm has been widely used as a learning algorithm for feedforward neural network training. In this paper, we prove a weak convergence theorem of an online gradient algorithm with a penalty term, assuming that the training examples are input in a stochastic way. The monotonicity of the error function in the iteration and the boundedness of the weight are both guaranteed. We also present a numerical experiment to support our results.     

Composition Operators on Orlicz–Lorentz Spaces

In this paper, we study the boundedness and the compactness of composition operators on Orlicz–Lorentz spaces.      

Transferring Boundedness from Conjugate Operators Associated with Jacobi, Laguerre, and Fourier–Bessel Expansions to Conjugate Operators in the Hankel Setting

In this paper we establish transference results showing that the boundedness of the conjugate operator associated with Hankel transforms on Lorentz spaces can be deduced from the corresponding boundedness of the conjugate operators defined on Laguerre, Jacobi, and Fourier–Bessel settings. Our result also allows us to characterize the power weights in order that conjugation associated with Laguerre, Jacobi, and Fourier–Bessel expansions define bounded operators between the corresponding weighted L ^p spaces. This paper is partially supported by MTM2004/05878. Third and fourth authors are also partially supported by grant PI042004/067.