Necessary and Sufficient Conditions for the <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Convergence of Double Fourier Series

We extend the results of paper of F.Móricz (2010), where necessary conditions were given for the L¹-convergence of double Fourier series. We also give necessary and sufficient conditions for the L¹-convergence under appropriate assumptions.     

Continuous selections for the metric projection onC 1

C ₁(K) is the space of real continuous functions onK endowed with the usualL ₁,-norm where \(K = \overline {\operatorname{int} K}\) is compact inR ^{m · U} is a finite-dimensional subspace ofC ₁,(K). The metric projection ofC ₁,(K) ontoU contains a continuous selection with respect toL ₁, -convergence if and only ifU is a unicity (Chebyshev) space forC ₁,(K). Furthermore, ifK is connected andU is not a unicity space forC ₁,(K), then there is no continuous selection with respect toL _∞-convergence. An example is given of aU and a disconnectedK with no continuous selection with respect toL ₁-convergence, but many continuous selections with respect toL _∞-convergence.     

Homogenization of unbounded singular integrals in W 1,∞

We study homogenization by ??-convergence, with respect to the L ¹-strong convergence, of periodic multiple integrals in W ^1,?? when the integrand can take infinite values outside of a convex bounded open set of matrices.     

Local and global uniform convergence for elliptic problems on varying domains

The aim of the paper is to prove optimal results on local and global uniform convergence of solutions to elliptic equations with Dirichlet boundary conditions on varying domains. We assume that the limit domain be stable in the sense of Keldyš [Amer. Math. Soc. Transl. 51 (1966) 1-73]. We further assume that the approaching domains satisfy a necessary condition in the inside of the limit domain, and only require L²-convergence outside. As a consequence, uniform and L²-convergence are the same in the trivial case of homogenisation of a perforated domain. We are also able to deal with certain cracking domains.     

A note on the weak invariance principle for local times

We prove uniform L²-convergence of local times of aperiodic recurrent random walks to local times of strictly α-stable processes with α > 1.     

An application of MVBV condition in real sense for L 1-convergence of trigonometric series

We continue to investigate the application of the Mean Value Bounded Variation (MVBV) concept in real sense, and prove the necessary and sufficient condition for L ¹-convergence of trigonometric series.     

Integrability and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-convergence of double cosine trigonometric series

We study here L1-convergence of new modified double cosine trigonometric sum and obtain a new necessary and sufficient condition for L1-convergence of double cosine trigonometric series.Also,the results obtained by Moricz[1],[2] are particular cases of ours.     

Integrability andL 1-convergence of sine series with generalized quasi-convex coefficients

In this paper we obtain a necessary and sufficient condition for the sine series with generalized quasi-convex coefficients to be a Fourier series. Also we studyL ¹-convergence of this series under the said condition on the coefficients.     

On integrability and L-convergence differentiated trigonometric series

New integrability and L¹-convergence classes for the r-times differentiated trigonometric series are derived using known results on integrability and L¹-convergence of trigonometric series. These classes, Theorems 1 and 2, subsume all known integrability and L¹-convergence classes for differentiated series. In particular, the extensions of the Fomin type theorems to r-times differentiated series, due to S.S. Bhatia and B. Ram (Proc. Amer. Math. Soc. 124 (1996) 1821-1829) and S.Y. Sheng (Proc. Amer. Math. Soc. 110 (1990) 895-904), are deduced from our Theorem 2 and its corollary Theorem 4. Another extension of the known results for differentiated series is given in Theorem 6. This result is proved using Fomin's theorem on cosine and sine series (Mat. Zametki 23 (1978) 213-222), and as a corollary of Theorem 2. The first proof yields another interesting conclusion, namely that the Fomin type theorems for differentiated series can be deduced as consequences of the original Fomin's results.     

Some limit theorems for negatively associated random variables

Let {X _n , n ≥ 1} be a sequence of negatively associated random variables. The aim of this paper is to establish some limit theorems of negatively associated sequence, which include the L ^p -convergence theorem and Marcinkiewicz–Zygmund strong law of large numbers. Furthermore, we consider the strong law of sums of order statistics, which are sampled from negatively associated random variables.     

Approximation and relaxation of perimeter in the Wiener space

We characterize the relaxation of the perimeter in an infinite dimensional Wiener space, with respect to the weak L²$L^2$-topology. We also show that the rescaled Allen–Cahn functionals approximate this relaxed functional in the sense of Γ-convergence.     

Estimates of derivatives ofn-convex functions

In the form of nonlinear norm estimates, a quantitative refinement of the following fact is given: for n-convex functions L¹-convergence implies Cⁿ⁻²-convergence. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 123–133. Translated by I. A. Fedortsova.     

On L 1-convergence of sine series

Our aim is to find the source why the logarithm sequences play the crucial role in the L ¹-convergence of sine series. We define three new classes of sequences; one of them has the character of the logarithm sequences, the other two are the extensions of the class defined by Zhou and named Logarithm Rest Bounded Variation Sequences. In terms of these classes, extended analogues of Zhou??s theorems are proved.     

First-order L1-convergence for relaxation approximations to conservation laws

We derive a first-order rate of L¹-convergence for stiff relaxation approximations to its equilibrium solutions, i.e., piecewise smooth entropy solutions with finitely many discontinuities for scalar, convex conservation laws. The piecewise smooth solutions include initial central rarefaction waves, initial shocks, possible spontaneous formation of shocks in a future time, and interactions of all these patterns. A rigorous analysis shows that the relaxation approximations to approach the piecewise smooth entropy solutions have L¹-error bound of O(ε|log ε| + ε), where ε is the stiff relaxation coefficient. The first-order L¹-convergence rate is an improvement on the error bound. If neither central rarefaction waves nor spontaneous shocks occur, the error bound is improved to O(ε). © 1998 John Wiley & Sons, Inc.     

ExponentialL 2-convergence andL 2-spectral gap for Markov processes

This paper deals with the exponentialL ²-convergence for jump processes. We introduce some reduction methods and improve some previous results. Then we prove that for birth-death processes, exponentialL ²-convergence coincides indeed with exponential ergodicity which is widely studied in the Markov chain theory. Research supported in part by the National Natural Science Foundation of China and Fok Ying-Tung Educational Foundation     

Existence ofBV absolute discontinuous minima for modified multiple integrals of the calculus of variations

We continue here the discussion on the existence of discontinuousBV minima, for a class of multiple integrals of the calculus of variationsI, we have started in [2] in view of possible studies on hyperbolic partial differential equations. Besides the associated Serrin integralJ, based onL ₁-convergence, we take into consideration a modified Serrin-type functionalJ ^*. This new integralJ ^* will be needed in [3] to prove Rankine-Hugoniot type properties.     

Approximations of Euler–Maruyama type for stochastic differential equations with Markovian switching,under non-Lipschitz conditions

We develop the Euler–Maruyama scheme for a class of stochastic differential equations with Markovian switching (SDEwMSs) under non-Lipschitz conditions  . Both L¹$L^1$ and L²$L^2$-convergence are discussed under different non-Lipschitz conditions. To overcome the mathematical difficulties arisen from the Markovian switching as well as the non-Lipschitz coefficients, several new analytical techniques have been developed in this paper which should prove to be very useful in the numerical analysis of stochastic systems.     

Coefficient Condition forL1-Convergence of Walsh–Fourier Series

In this paper we give a condition with respect to Walsh–Fourier coefficients that implies theL₁-convergence of the corresponding Walsh–Fourier series. We show that theL₁-convergence class induced by this condition contains each one of the previously known convergence classes as a proper subset. We also show that our condition implies not only theL₁-convergence but also the convergence in the dyadic Hardy norm if the function represented by the series belongs to the dyadic Hardy space.     

On Lp-convergence of Grunwald interpolation

In this paper, the L_p-convergence of Grünwald interpolation G_n(f,x) based on the zeros of Jacobi polynomials J _n ^(α,β) (x)(−1<α,β<1) is considered. L_p-convergence (0<p<2) of Grünwald interpolation G_n(f,x) is proved for p·Max(α,β)<1. Moreover, L_p-convergence (p>0) of G_n(f,x) is obtained for −1<α,β≤0. Therefore, the results of [1] and [3–5] are improved.     

On <Emphasis Type="Italic">L</Emphasis> <Subscript>p</Subscript>-convergence of Cesàro means for Fourier series with monotonic coefficients

For sine and cosine Fourier serieswithmonotonic coefficients we study L_p-convergence (1 < p < ∞) of their Cesàro means of a negative order.