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 .     

级数的常规可和，Cesàro可和与Abel可和的几点讨论

讨论级数常规可和、Cesaro可和与Abel可和的关系．利用数学分析级数理论,证明Abel可和适用范围最广,Cesaro可和其次,级数常规可和适用范围最小．这个结论丰富了经典级数理论,为实际应用中选用合适可和提供依据．     

Characterizations of completeness of normed spaces through weakly unconditionally Cauchy series

In this paper we obtain two new characterizations of completeness of a normed space through the behaviour of its weakly unconditionally Cauchy series. We also prove that barrelledness of a normed space X can be characterized through the behaviour of its weakly-* unconditionally Cauchy series in X*.     

On general summability factor theorem of infinite series

A lower triangular matrix with nonzero principal diagonal entries is called a triangle. In this paper we obtain the sufficient conditions for ∑a_nλ_n to be summable ∣A∣_k whenever ∑a_n is summable ∣T∣_k for a triangle T.     

On absolute summability factors of infinite series

In this paper using δ-quasi-monotone sequences a theorem on summability factors of infinite series, which generalizes a theorem of Bor [4] on summability factors of infinite series, is proved. Also, in the special case this theorem includes a result of Mazhar [8] on |C, 1|_k summability factors.     

On application of matrix summability to Fourier series

Bor has recently obtained a main theorem dealing with absolute weighted mean summability of Fourier series. In this paper, we generalized that theorem for summability method. Also, some new and known results are obtained dealing with some basic summability methods.     

Multipliers for the absolute euler summability of fourier series

In this paper, the author has investigated necessary and sufficient conditions for the absolute Euler summability of the Fourier series with miltipliers. These conditions are weaker than those obtained earlier by some workers. It is further shown that the multipliers are best possible in certain sense.     

Matrix characterization of A-statistical convergence

The main objective of the paper is to characterize multipliers of summability fields of regular methods, while relaxing the usual boundedness condition. For this purpose we use the sequence spaces of A-bounded sequences and A-uniformly integrable sequences. Among the main results, it is shown that the space of multipliers is closely related to the space of A-statistically convergent sequences and that A-statistical convergence over ?_∞ is equivalent to a regular matrix method. This observation eliminates the need for separate proofs of several A-statistical convergence results.     

Generalized absolute Cesaro summability of a Fourier series

In an attempt to study the scope of a theorem due to Pati, the authors have established that φ(t) logK|t∈B _u V in (0,π)⟹ΣA _n (x) is |C, 0,β| forβ>1, at the pointt = x.     

Extension on absolute summability factors of infinite series

Extension and improvement has been made on a recent result proved by Mazhar [S.M. Mazhar, Absolute summability factors of infinite series, Kyungpook Math. J. 39 (1999) 67-73] concerning absolute summability factors of infinite series.     

Absolute matrix summability of the allied series of a Fourier series

In this paper we have proved a theorem for infinite triangular matrix using new techniques.     

Factors for ¦N,pn;δ¦k summability of Fourier series

In this paper two theorems on | N,p_n;δ|_k summability factors, which generalize the results of Bor [4] on | N,p_n|k summability factors, have been proved.     

On the|\bar N,p_n | summability factors of infinite seriessummability factors of infinite series

In this paper a theorem of Chen [1] on summability factors of infinite series has been proved under weaker conditions.     

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.     

On the uniform nonsingularity of matrices of search directions and the rate of convergence in minimization algorithms

A convergent minimization algorithm made up of repetitive line searches is considered in ⁿ . It is shown that the uniform nonsingularity of the matrices consisting ofn successive normalized search directions guarantees a speed of convergence which is at leastn-step Q-linear. Consequences are given for multistep methods, including Powell's 1964 procedure for function minimization without calculating derivatives as well as Zangwill's modifications of this procedure.The authors wish to thank the Namur Department of Mathematics, especially its optimization group, for many discussions and encouragement. They also thank the reviewers for many helpful suggestions.     

Unconditionally Cauchy series and Cesàro summability

In this paper we obtain new characterizations of weakly unconditionally Cauchy series and unconditionally convergent series through Cesàro summability. We study new spaces associated to a series in a Banach space; as a consequence, new characterizations of complete and barrelled normed spaces are proved.     

Unconditionally Converging Multilinear Operators

We introduce a notion of unconditionally converging multilinear operator which allows to extend many of the results of the linear case to the multilinear case. We prove several characterizations of these multilinear operators (one of which seems to be new also in the linear case), which allow to considerably simplify the work with this kind of operators.     

On the uniform convergence of trigonometric series

In this paper, we obtain sufficient conditions for the uniform convergence of trigonometric series with monotone (in the extended sense) coefficients.