The deviation of polygonal functions in the Lp metric

The precise value is given of the upper bound of the deviation in the L_p metric (1 < p < ) of a function f(x) in the class H , given by a convex modulus of continuity(t), from its polygonal approximation at the points x_k=k/n (k=0, 1 ...,n).Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 31–37, 1969.I would like to express my appreciation to A. A. Nudel'man for suggesting the problem considered here and for his help.     

Estimation of the convergence rate in a transfer theorem for centered random sums

Estimates of the convergence rate in a transfer theorem for random sums centered by constants are presented. The estimates are obtained as a consequence of the estimates of the stability of the distributions of the sums represented in the form of special mixtures to small variations in the mixing and miscible distributions.     

On the rate of superlinear convergence of a class of variable metric methods

Summary This paper considers a class of variable metric methods for unconstrained minimization. Without requiring exact line searches each algorithm in this class converges globally and superlinearly on convex functions. Various results on the rate of the superlinear convergence are obtained.Dedicated to Professor Dr. H. Görtler on the occasion of his seventieth birthday     

Flat convergence for integral currents in metric spaces

In compact local Lipschitz neighborhood retracts in weak convergence for integral currents is equivalent to convergence with respect to the flat distance. This comes as a consequence of the deformation theorem for currents in Euclidean space. Working in the setting of metric integral currents (the theory of which was developed by Ambrosio and Kirchheim) we prove that the equivalence of weak and flat convergence remains true in the more general context of metric spaces admitting local cone type inequalities. These include in particular all Banach spaces and all CAT(κ)-spaces. As an application we obtain the existence of a minimal element in a fixed homology class and show that the weak limit of a sequence of minimizers is itself a minimizer.     

The best approximation of the differentiation operator in the metric of Lp

For Stechkin's problem of the best approximation for the differentiation operator we indicate the necessary and sufficient conditions that E_n be finite. We study some properties of continuous linear operators V from L_p into L_q.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 531–538, November, 1972.     

On convergence in fuzzy metric spaces

The concept of p-convergence in fuzzy metric spaces, in George and Veeramani's sense, has been recently given by D. Mihet in [D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems 158 (2007) 915-921]. In this note we study some aspects relative to this concept and characterize those fuzzy metric spaces, that we call principal, in which the family of p-convergent sequences agrees with the family of convergent sequences. Also a non-completable fuzzy metric space, which is not principal, is given.     

Strong ideal convergence in probabilistic metric spaces

In the present paper we introduce the concepts of strongly ideal convergent sequence and strong ideal Cauchy sequence in a probabilistic metric (PM) space endowed with the strong topology, and establish some basic facts. Next, we define the strong ideal limit points and the strong ideal cluster points of a sequence in this space and investigate some properties of these concepts.