On mean convergence of quasi-Hermite-interpolation

In this paper, the mean convergence of the quasi-Hermite-Fejér interpolation R_n(f,x) and the mean conergence of the quasi-Hermite interpolation R _n ^* (f,x) and its derivative based on the zeros of (1−x²)P_n (where P_n(x) be the Legendre polynomial) are respectively considered. The degree of corresponding L₂-approximation is given.     

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.     

On the convergence in mean of trigonometric Fourier series

We prove the sharpness of Zygmund’s theorem, which asserts that if a 2π-periodic function f belongs to L ln⁺ L, then its Fourier series is convergent in mean.     

Remarks on mean convergence

In this article a necessary and sufficient condition is established for the validity of the Cauchy criterion in the class of sequences of functions {f_n} of the set (L) which are convergent with respect to the -distance.Translated from Matematicheskie Zametki, Vol.Translated from Matematicheskie Zametki, Vol.     

On the mean value of certain functions connected with the convergence of orthogonal series

В статье рассматрива ются множестваQ ⁿ , 1≦п<∞, ортонормированных с истемΦ={φ _i (x)} _i ⁿ =1, состоящих из функций, постоянных на интервалах \(\left( {\frac{{j - 1}}{n}, \frac{j}{n}} \right)\) , 1 ≦j ≦j ≦п. НаQ ⁿ естественно перенос ится с группы ортогон альных матриц порядкаn мера Хаара. Изучается поведение наQ ⁿ функци и $$S(\Phi ) = \mathop {\sup }\limits_{\mathop \sum \limits_{i = 1}^n y_i^2 = 1} (\int\limits_0^1 {\mathop {sup}\limits_{1 \leqq r \leqq n} } (\mathop \sum \limits_{i = 1}^n y_i \varphi (x))^2 dx)^{1/2} $$ . Доказывается, что приt > 0 иn=1,2,... $$\mu \{ \Phi \in Q^n :s(\Phi ) \geqq t\} \leqq (Ce^{ - \gamma t^2 } )^n $$ .     

Rate of convergence of the mean curvature flow

We study the flow M_t of a smooth, strictly convex hypersurface by its mean curvature in ?^{n + 1}. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time T and point x^* (which is due to Huisken). This is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere Sⁿ of radius √n. In this paper we will study the rate of exponential convergence of a rescaled flow. We will present here a method that tells us that the rate of the exponential decay is at least 2/n. We can define the “arrival time” u of a smooth, strictly convex, n‐dimensional hypersurface as it moves with normal velocity equal to its mean curvature via u(x) = t if x ∈ M_t for x ∈ Int(M₀). Huisken proved that, for n ≥ 2, u(x) is C² near x^*. The case n = 1 has been treated by Kohn and Serfaty [11]; they proved C³‐regularity of u. As a consequence of the obtained rate of convergence of the mean curvature flow, we prove that u is not necessarily C³ near x^* for n ≥ 2. We also show that the obtained rate of convergence 2/n, which arises from linearizing a mean curvature flow, is the optimal one, at least for n ≥ 2. © 2007 Wiley Periodicals, Inc.     

On rates of convergence for sample average approximations in the almost sure sense and in mean

Mathematical Programming - We study the rates at which optimal estimators in the sample average approximation approach converge to their deterministic counterparts in the almost sure sense and in...     

On the convergence of an approximate deconvolution model to the 3D mean Boussinesq equations

In this paper, we study an LES model for the approximation of large scales of the 3D Boussinesq equations. This model is obtained using the approach first described by Stolz and Adams, based on the Van Cittern approximate deconvolution operators, and applied to the filtered Boussinesq equations. Existence and uniqueness of a regular weak solution are provided. Our main objective is to prove that this solution converges towards a solution of the filtered Boussinesq equations, as the deconvolution parameter goes to zero. Copyright © 2014 John Wiley & Sons, Ltd.     

On complete convergence in mean for double sums of independent random elements in Banach spaces

Conditions are provided under which a normed double sum of independent random elements in a real separable Rademacher type p Banach space converges completely to 0 in mean of order p. These conditions for the complete convergence in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. In case the Banach space is not of Rademacher type p, it is proved that the complete convergence in mean of order p of a normed double sum implies a strong law of large numbers.     

Some mean convergence and complete convergence theorems for sequences of m-linearly negative quadrant dependent random variables

The structure of linearly negative quadrant dependent random variables is extended by introducing the structure of m-linearly negative quadrant dependent random variables (m = 1, 2, …). For a sequence of m-linearly negative quadrant dependent random variables {X _{n, n} ? 1} and 1 < p < 2 (resp. 1 ? p < 2), conditions are provided under which $n^{ - 1/p} \sum\limits_{k = 1}^n {\left( {\left. {X_k - } \right|EX_k } \right) \to } 0$ in L ¹. Moreover, for 1 ? p < 2, conditions are provided under which $n^{ - 1/p} \sum\limits_{k = 1}^n {\left( {X_k - EX_k } \right)}$ converges completely to 0. The current work extends some results of Pyke and Root (1968) and it extends and improves some results of Wu, Wang, and Wu (2006). An open problem is posed.     

A convergence proof for linear mean value cross decomposition

The mean value cross decomposition method for linear programming problems is a modification of ordinary cross decomposition that eliminates the need for using the Benders or Dantzig-Wolfe master problem. It is a generalization of the Brown-Robinson method for a finite matrix game and can also be considered as a generalization of the Kornai-Liptak method. It is based on the subproblem phase in cross decomposition, where we iterate between the dual subproblem and the primal subproblem. As input to the dual subproblem we use the average of a part of all dual solutions of the primal subproblem, and as input to the primal subproblem we use the average of a part of all primal solutions of the dual subproblem. In this paper we give a new proof of convergence for this procedure. Previously convergence has only been shown for the application to a special separable case (which covers the Kornai-Liptak method), by showing equivalence to the Brown-Robinson method.     

Notes on convergence of geometric mean for fuzzy numbers

In this paper, we give some sufficient conditions for convergence of geometric mean forT-related fuzzy numbers whereT is an Archimedeant-norm under some mild restrictions. These results generalize earlier results of Hong and Lee [Fuzzy Sets and Systems 116 (2000) 263–267].     

Weighted mean convergence of Hakopian interpolation on the disk

In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn(f;x,y) and a smooth function g(x,y) on D converges to that of f(x,y) and g(x,y) on D when n →∞, provided f(x,y) belongs to C(D) and all first partial derivatives of g(x,y) belong to the space LipαM(0 ＜α≤ 1). We further show that provided all second partial derivatives of g(x,y) also belong to the space LipαM and f(x,y) belongs to C1 (D), the inner product between the partial derivative of Hakopian interpolation polynomial (6)/(6)xHn(f;x,y) and g(x,y) on D converges to that between (6)/(6)xf(x,y) and g(x,y) on D when n →∞.     

Estimate for evolutionary surfaces of prescribed mean curvature and the convergence

In the paper, we will discuss the gradient estimate for the evolutionary surfaces of prescribed mean curvature with Neumann boundary value under the condition $f_\tau\ge -\kappa$, which is the same as the one in the interior estimate by K. Ecker and generalizes the condition $f_\tau\ge 0$ studied by Gerhardt etc. Also, based on the elliptic result obtained recently, we will show the longtime behavior of surfaces moving by the velocity being equal to the mean curvature.