[en] The paper is concerned with the behaviour of the coefficients of multiple Walsh-Paley series that are cube convergent to a finite sum. It is shown that even an everywhere convergent series of this kind may contain coefficients with numbers from a sufficiently large set that grow faster than any preassigned sequence. By Cohen's theorem, this sort of thing cannot happen for multiple trigonometric series that are cube convergent on a set of full measure — their coefficients cannot grow even exponentially. Null subsequences of coefficients are determined for multiple Walsh-Paley series that are cube convergent on a set of definite measure. Bibliography: 18 titles.

[en] In the present work, we define a new type of statistical convergence by using the notion of the relatively uniform convergence. We prove a Korovkin-type approximation theorem with the help of this new definition. Then, we construct a strong example that satisfies our theory. Finally, we compute the rate of statistical relatively equal convergence.

[en] We investigate problems on a.e. convergence of Riemann sums R_nf(x)=1/n Σ_k=0^n-1f(x+k/n), x element of T, with the use of classical maximal functions in Rⁿ. A theorem on the equivalence of Riemann and ordinary maximal functions is proved, which allows us to use techniques and results of the theory of differentiation of integrals in Rⁿ in these problems. Using this method we prove that for a certain sequence (n_k) the Riemann sums R_nkf(x) converge a.e. to f element of L^p, p>1. Bibliography: 23 titles.

[en] Classical information that is retrieved from a quantum tetrahedron is intrinsically fuzzy. We present an asymptotically optimal generalized measurement for its extraction. For a single tetrahedron the optimal uncertainty in dihedral angles is shown to scale as an inverse of the surface area. The introduced commutative observables allow us to demonstrate how the clustering of many small tetrahedra leads to a faster convergence to a classical geometry.

[en] We present a local convergence analysis of a two-step and derivative-free Kung–Traub’s method, which is based on a parameter and has fourth order of convergence. Using basins of attraction of the method, dynamical behavior of the scheme is studied and the best choice of the parameter is found in the sense of reliability and stability. Some illustrative examples show that as the parameter gets close to zero, radius of convergence of the method becomes larger.

[en] In this paper, the concepts of weighted statistical pointwise convergence, weighted statistical uniform convergence and weighted equi-statistical convergence are introduced. Then, using weighted equi-statistical convergence, a general Korovkin type theorem is obtained. Also, an example such that our new approximation result works but its classical case does not work is constructed.

[en] In this paper we develop a self-adaptive projection and contraction method for the linear complementarity problem (LCP). This method improves the practical performance of the modified projection and contraction method by adopting a self-adaptive technique. The global convergence of our new method is proved under mild assumptions. Our numerical tests clearly demonstrate the necessity and effectiveness of our proposed method

[en] Based on the notion of the ε -subgradient, we present a unified technique to establish convergence properties of several methods for nonsmooth convex minimization problems. Starting from the technical results, we obtain the global convergence of: (i) the variable metric proximal methods presented by Bonnans, Gilbert, Lemarechal, and Sagastizabal, (ii) some algorithms proposed by Correa and Lemarechal, and (iii) the proximal point algorithm given by Rockafellar. In particular, we prove that the Rockafellar-Todd phenomenon does not occur for each of the above mentioned methods. Moreover, we explore the convergence rate of { parallel x_k parallel } and {f(x_k) } when {x_k } is unbounded and {f(x_k) } is bounded for the non-smooth minimization methods (i), (ii), and (iii)

[en] A necessary and sufficient condition over a sequence {A_n}_n is an element of N of σ-subalgebras that assures L^p-convergence of the conditional expectations will be given. The result generalizes the L^p-martingales, O'Reilly-Fetter and Boylan (equiconvergence) theorems. (author). 5 refs

[en] Let $\stackrel{^<!--\; ^\; -->}{\mathrm{\sigma <!--\; ??\; -->}}$ be a Cauchy transform of a possibly complex-valued Borel measure $\mathrm{\sigma <!--\; ??\; -->}$ and $\{p_n\}$ a system of orthonormal polynomials with respect to a measure $\mathrm{\mu <!--\; ??\; -->}$, where $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathrm{\mu <!--\; ??\; -->})\cap <!--\; ???\; -->\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathrm{\sigma <!--\; ??\; -->})=\mathrm{\varnothing <!--\; ???\; -->}$. An $(m,n)$th Frobenius-Padé approximant to $\stackrel{^<!--\; ^\; -->}{\mathrm{\sigma <!--\; ??\; -->}}$ is a rational function $P/Q$, $\mathrm{deg}<!--\; ???\; -->(P)\le <!--\; ???\; -->m$, $\mathrm{deg}<!--\; ???\; -->(Q)\le <!--\; ???\; -->n$, such that the first $m+n+1$ Fourier coefficients of the remainder function $Q\stackrel{^<!--\; ^\; -->}{\mathrm{\sigma <!--\; ??\; -->}}-<!--\; ???\; -->P$ vanish when the form is developed into a series with respect to the polynomials $p_n$. We investigate the convergence of the Frobenius-Padé approximants to $\stackrel{^<!--\; ^\; -->}{\mathrm{\sigma <!--\; ??\; -->}}$ along ray sequences $n/(n+m+1)\to <!--\; ???\; -->c>0$, $n-<!--\; ???\; -->1\le <!--\; ???\; -->m$, when $\mathrm{\mu <!--\; ??\; -->}$ and $\mathrm{\sigma <!--\; ??\; -->}$ are supported on intervals of the real line and their Radon-Nikodym derivatives with respect to the arcsine distribution of the corresponding interval are holomorphic functions.

Bibliography: 30 titles. (paper)