A Note on the Convergence of Orlicz-Bochner Spaces with the Luxemburg Norm

In this paper, the convergence of the Orlicz-Bochner function spaces with the Luxemburg norm was proved.

WEAK ORLICZ SPACE AND ITS CONVERGENCE THEOREMS

In this article, a class of weak Orlicz function spaces is defined and their basic properties are discused. In particular, for the sequences in weak Orlicz space, we establish several basic convergence theorems including bounded convergence theorem, control convergence theorem and Vitali-type convergence theorem and so on. Moreover, the conditional compactness of its subsets is also discussed.

QUASI-WEAK CONVERGENCE WITH APPLICATIONS IN ORDERED BANACH SPACE

In the paper quasi_weak convergence is introduced in ordered Banach space and it is weaker than weak convergence. Besed on it, the fixed point existence theorem of increasing operator is proved without the suppose of continuity and compactness in the sense of norm and weak compactness and is applied to the Hammerstein nonlinear intergal equation.

CONVERGENCE OF DOUBLE WALSH-FOURIER SERIES AND HARDY SPACES

It is proved that the maximal operator of the Marczinkiewicz-Fejér meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space H p to L p (2/3<p<∞) and is of weak type (1,1). As a consequence we obtain that the Marczinkiewicz-Fejér means of a function f∈L 1 converge a.e. to the function in question. Moreover, we prove that these means are uniformly bounded on H p whenever 2/3<p<∞. Thus, in case f∈H p , the Marczinkiewicz-Fejér means conv f in H p norm. The same results are proved for the conjugate means, too.

Notes on Some Convergences in Riesz Space

An equivalent description of u-uniform convergence is presented first. Then the relations among the order convergence, u-uniform convergence and norm convergence of sequences are discussed in Riesz spaces. An equivalence of the three convergences is brought forward; namely, {fn} is a u-uniform Cauchy sequence. Finally the relations among the three convergences of sequences are also extended to the relations among the convergences of nets in Riesz spaces.

Convergence Rates of Density Estimation in Besov Spaces

The optimality of a density estimation on Besov spaces Bsr,q(R) for the Lp risk was established by Donoho, Johnstone, Kerkyacharian and Picard (“Density estimation by wavelet thresholding,” The Annals of Statistics, Vol. 24, No. 2, 1996, pp. 508-539.). To show the lower bound of optimal rates of convergence Rn(Bsr,q, p), they use Korostelev and Assouad lemmas. However, the conditions of those two lemmas are difficult to be verified. This paper aims to give another proof for that bound by using Fano’s Lemma, which looks a little simpler. In addition, our method can be used in many other statistical models for lower bounds of estimations.

A Monotonicity Condition for Strong Convergence of the Mann Iterative Sequence for Demicontractive Maps in Hilbert Spaces

Let?be a real Hilbert space and?C?be a nonempty closed convex subset of H. Let T : C?→?C?be a demicontractive map satisfying?〈Tx, x〉?≥?‖x‖2 for all?x?∈ D (T). Then the Mann iterative sequence given by?xn + 1?= (1 - an) xn +?anT xn, where an ∈?(0, 1) n?≥?0, converges strongly to an element of F (T):= {x?∈ C : Tx = x}. This strong convergence is obtained without the compactness-type assumptions on C, which many previous results (see e.g. [1]) employed.

The rate of convergence on fractional power dissipative operator on some sobolev type spaces

In[3],Chen,Deng,Ding and Fan proved that the fractional power dissipative operator is bounded on Lebesgue spaces Lp(Rn),Hardy spaces Hp(Rn)and general mixed norm spaces,which implies almost everywhere convergence of such operator.In this paper,we study the rate of convergence on fractional power dissipative operator on some sobolev type spaces.

CONVERGENCE RATES FOR LACUNARY TRIGONOMETRIC INTERPOLATION IN _(2π)~p SPACES

The saturation rate and class of (0,m1,m2, …,mq) trigonometric inter polation operators in . spaces have been determined by Cavaretta and Selvaraj. In this paper, we consider the convergence and saturation problems of these operators in (1≤p≤∞) and obtain complete results.

COMPLETE MOMENT CONVERGENCE FOR L^P-MIXINGALES

In this paper, the complete moment convergence for L~p-mixingales are studied.Sufficient conditions are given for the complete moment convergence for the maximal partial sums of B-valued L~p-mixingales by utilizing the Rosenthal maximal type inequality for B-valued martingale difference sequence, which extend and improve the related known works in the literature.

ON WEAK FILTER CONVERGENCE AND THE RADON-RIESZ TYPE THEOREM

The author shows a characterization of a （unbounded） weakly filter convergent sequence which is parallel to that every weakly null sequence （xn） in a Banach space admits a norm null sequence （yn） with yn ∈ co（xk）k≥n for all n ∈ N. A version of the Radon-Riesz type theorem is also proved within the frame of the filter convergence.

Convergence analysis for the Secant method based on new recurrence relations

A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.

Weak Convergence Theorems for Nonself Mappings

Let E be a real uniformly convex and smooth Banach space, and K be a nonempty closed convex subset of E with P as a sunny nonexpansive retraction. Let T1, T2 ： K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with a sequence {kn^（i）} [1, ∞） （i = 1, 2）, and F ：= F（T1）∩ F（T2） ≠ 0. An iterative sequence for approximation common fixed points of the two nonself asymptotically nonexpansive mappings is discussed. If E has also a Frechet differentiable norm or its dual E^＊ has Kadec-Klee property, then weak convergence theorems are obtained.

An Efficient Second-Order Convergent Scheme for One-Side Space Fractional Diffusion Equations with Variable Coefficients

In this paper,a second-order fnite-diference scheme is investigated for time-dependent space fractional difusion equations with variable coefcients.In the presented scheme,the Crank-Nicolson temporal discretization and a second-order weighted-and-shifted Grünwald-Letnikov spatial discretization are employed.Theoretically,the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the variable coefcients.Moreover,a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme.The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes,so that the Krylov subspace solver for the preconditioned linear systems converges linearly.Numerical results are reported to show the convergence rate and the efciency of the proposed scheme.

Characterizing Continuous dcpos by Liminf Convergence of Filters

It is proved in this note that,under a mild assumption,a dcpo L is continuous if and only if the liminf convergence on L is topological.

ON ALMOST SURE CONVERGENCE OF WEIGHTED SUMS OF RANDOM ELEMENT SEQUENCES

We mainly study the almost sure limiting behavior of weighted sums of the form ∑ni=1 aiXi/bn , where {Xn, n ≥ 1} is an arbitrary Banach space valued random element sequence or Banach space valued martingale difference sequence and {an, n ≥ 1} and {bn,n ≥ 1} are two sequences of positive constants. Some new strong laws of large numbers for such weighted sums are proved under mild conditions.

CONVERGENCE RATES IN THE LAW OF LARGE NUMBERS FOR B-VALUED RANDOM ELEMENTS

The author discusses necessary and sufficient conditions of the complete con- vergence for sums of B-valued independent but not necessarily identically distributed r.v.'s in Banach space of type p, and obtains characterization of Banach space of type p in terms of the complete convergence. A series of classical results on iid real valued r.v.'s are ex- tended. As application authors give the analogous results for randomly indexed sums.

CONVERGENCE OF HYBRID VISCOSITY AND STEEPEST-DESCENT METHODS FOR PSEUDOCONTRACTIVE MAPPINGS AND NONLINEAR HAMMERSTEIN EQUATIONS

In this article, we first introduce an iterative method based on the hybrid viscos- ity approximation method and the hybrid steepest-descent method for finding a fixed point of a Lipschitz pseudocontractive mapping （assuming existence） and prove that our proposed scheme has strong convergence under some mild conditions imposed on algorithm parameters in real Hilbert spaces. Next, we introduce a new iterative method for a solution of a non- linear integral equation of Hammerstein type and obtain strong convergence in real Hilbert spaces. Our results presented in this article generalize and extend the corresponding results on Lipschitz pseudocontractive mapping and nonlinear integral equation of Hammerstein type reported by some authors recently. We compare our iterative scheme numerically with other iterative scheme for solving non-linear integral equation of Hammerstein type to verify the efficiency and implementation of our new method.

DETERMINATION OF SPACE GROUP OF PrCo_(12)B_6 COMPOUND BY CONVERGENT BEAM ELECTRON DIFFRACTION

The space group of PrCo_(12)B_6,compound has been determined using the convergent beam elec- tron diffraction method.The space group is found to be R3m.

THE SPACES OF CESRO ALMOST CONVERGENT SEQUENCES AND CORE THEOREMS

As known, the method to obtain a sequence space by using convergence field of an infinite matrix is an old method in the theory of sequence spaces. However, the study of convergence field of an infinite matrix in the space of almost convergent sequences is so new （see [15]）. The purpose of this paper is to introduce the new spaces ^ ～f and fo consisting of all sequences whose Ceshro transforms of order one are in the spaces f and ^ ～ f0, respectively. Also, in this paper, we show that ^ ～f and ^ ～f0 are linearly isomorphic to the spaces f and f0, respectively. The β- and γ-duals of the spaces ^ ～f and 2% are computed. Furthermore, the classes （^ ～f： μ） and （μ ： f） of infinite matrices are characterized for any given sequence space μ, and determined the necessary and sufficient conditions on a matrix A to satisfy Bc-core（Ax） K-core（x）, K-core（Ax） Bg-core（x）, Bc-core（Ax） Be-core（x）, Bc-core（Ax） t-core（x） for all x ∈ t∞.