A generalization of almost convergence

In this paper, we discuss almost convergent sequences. The concept of almost convergence was first introduced by G. G. Lorentz in 1948. Since then, various types of generalizations for almost convergence have been constructed and studied. Our concern is also to generalize the almost convergence. We construct a generalized almost convergence (GAC) in a natural way through the use of Lorentz-type definition, and give an analogue of Lorentz’s result. It should be noted that the definition of our GAC does not require normed space nor boundedness of sequences, although Lorentz-type definitions usually require it. On the other hand, we also show that our GAC eventually has a certain type of boundedness in spite of its definition.     

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.     

Tauberian conditions for w-almost convergent double sequences

In this paper, we introduce the concept of w-almost convergent sequences. Such a definition is a weak form of almost convergent sequences given by G. G. Lorentz in [Acta Math. 80(1948),167-190]. We give a detailed study on w-almost convergent double sequences and prove that w-almost convergence and almost convergence are equivalent under the boundedness of the given sequence. The Tauberian results for w-almost convergence are established. Our Tauberian results generalize a result of Lorentz and Tauber’s second theorem. Moreover, we prove that w-almost convergence and norm convergence are equivalent for the sequence of the rectangular partial sums of the Fourier series of f ∈ L^p(T²), where 1 < p < ∞.      

MULTIPLIERS AND RELATIVE COMPLETION IN WEIGHTED LORENTZ SPACES

Let G be a locally compact Abelian group with Haar measure μ. In the present paper, first the authors discussed some properties of weighted Lorentz space. Then they defined the relative completion A of a subspace A of the weighted Lorentz space, and showed that the space of the multipliers from L_w~1(G) to A is algebrically isomorphic and homeomorphic to A.     

Uniform approximation by combinations of Bernstein polynomials

We prove a pointwise characterization result for combinations of Bernstein polynomials. The main result of this paper includes an equivalence theorem of H. Berens and G. G. Lorentz as a special case.     

Local rotundity structure of generalized Orlicz–Lorentz sequence spaces

We give some criteria for extreme points and strong U-points in generalized Orlicz–Lorentz sequence spaces, which were introduced in [P. Foralewski, H. Hudzik, L. Szymaszkiewicz, On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces, Math. Nachr. (in press)] (cf. [G.G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly 60 (1953) 176–179; M. Nawrocki, The Mackey topology of some F-spaces, Ph.D. Dissertation, Adam Mickiewicz University, Poznań, 1984 (in Polish)]). Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. This paper is related to the results from [A. Kamińska, Extreme points in Orlicz–Lorentz spaces, Arch. Math. 55 (1990) 173–180] (see Remark 1).     

MULTIPLIERS AND TENSOR PRODUCTS OF L（p, q） LORENTZ SPACES

Let G be a locally compact abelian group. The main purpose of this article is to find the space of multipliers from the Lorentz space L(p1, q1)(G) to L(p21,q21)(G). For this reason, the authors define the space Ap1,q1p2,q2(G), discuss its properties and prove that the space of multipliers from L(p1,q1)(G) to L(p21,q21)(G) is isometrically isomorphic to the dual of Ap1,q1p2q2(G).     

Kronecker-Ausdruck und Kreisverfahren der Limitierungstheorie

Ohne ZusammenfassungHerrn George G. Lorentz zum 60. Geburtstag am 25. Februar 1970 gewidmet     

Convexity and concavity constants in Lorentz and Marcinkiewicz spaces

We provide here the formulas for the q-convexity and q-concavity constants for function and sequence Lorentz spaces associated to either decreasing or increasing weights. It yields also the formula for the q-convexity constants in function and sequence Marcinkiewicz spaces. In this paper we extent and enhance the results from [G.J.O. Jameson, The q-concavity constants of Lorentz sequence spaces and related inequalities, Math. Z. 227 (1998) 129-142] and [A. Kamińska, A.M. Parrish, The q-concavity and q-convexity constants in Lorentz spaces, in: Banach Spaces and Their Applications in Analysis, Conference in Honor of Nigel Kalton, May 2006, Walter de Gruyter, Berlin, 2007, pp. 357-373].     

ON THE LORENTZ CONJECTURES UNDER THE L_1-NORM

Let f (x) ∈ C [-1, 1], p_n~* (x) be the best approximation polynomial of degree n tof (x). G. Iorentz conjectured that if for all n, p_(2n)~* (x) = p_(2n+1)~* (x), then f is even; and ifp_(2n+1)~* (x) = p_(2n+2)~* (x), p_o~* (z) = 0, then f is odd. In this paper, it is proved that, under the L_1-norm, the Lorentz conjecture is validconditionally, i. e. if (i) (1-x~2) f (x) can be extended to an absolutely convergentTehebyshev sories; (ii) for every n, f (x) - p_(2n+1)~* (x) has exactly 2n + 2 zeros (or, in thearcond situation, f (x) - p_(2n+2)~* (x) has exaetly 2n+3 zeros), then Lorentz conjecture isvalid.     

Tauberian conditions for almost convergence

In [Acta Math. 80(1948), 167–190], G. G. Lorentz characterized almost convergent sequences in (or in ) in terms of the concept of uniform convergence of the de la Vallée-Poussin means. In this paper, we present Tauberian results which relate almost convergence to norm convergence or to the (C, 1) convergence. Our results generalize Kronecker lemma. As a consequence, we prove that almost convergence and norm convergence are equivalent for the sequence of the partial sums of the Fourier series of (or ), where . We also show that our results can be used to derive Fatou’s theorem.      

Finite-Energy Global Well-Posedness of the Maxwell–Klein–Gordon System in Lorenz Gauge

It is known that the Maxwell–Klein–Gordon system (M–K–G), when written relative to the Coulomb gauge, is globally well-posed for finite-energy initial data. This result, due to Klainerman and Machedon, relies crucially on the null structure of the main bilinear terms of M–K–G in Coulomb gauge. It appears to have been believed that such a structure is not present in Lorenz gauge, but we prove here that it is, and we use this fact to prove finite-energy global well-posedness in Lorenz gauge. The latter has the advantage, compared to Coulomb gauge, of being Lorentz invariant, hence M–K–G in Lorenz gauge is a system of nonlinear wave equations, whereas in Coulomb gauge the system has a less symmetric form, as it contains also an elliptic equation.     

A. I. Mal'tsev's problem on operations on groups

In response to A. I. Mal'tsev's problem (1948), associative operations are constructed on the class of all groups; these operations are distinct from the direct and free products and are hereditary for subgroups of the factors.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 225–249, 1989.     

Active control of cylinder wake

The objective of this paper is to develop an efficient active control algorithm for manipulating wake flows past a solid cylinder in an electrically low-conducting fluid (e.g. seawater). The intent is to avoid both vortex shedding and flow separation from the body. It is expected to reduce the mean drag significantly. This is achieved through the introduction of a Lorentz force in the azimuthal direction generated by an array of permanent magnets and electrodes located on the solid structure. With the use of a symmetric and static Lorentz force over the entire surface of the cylinder, the vortex shedding behind the cylinder weakens and eventually disappears completely when the Lorentz force is sufficiently large. The localized Lorentz force along the rear surface of the cylinder was also used to control the vortex shedding behind the cylinder. In this case, numerical results show that the efficiency of the localized Lorentz force in controlling the flow is to that of the Lorentz force distributed over the whole surface.     

Mixed Norm and Multidimensional Lorentz Spaces

In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved ([16], [7]). However, the question for multidimensional Lorentz spaces is still open. In this paper, we consider weights of product type, and give necessary and sufficient conditions for the Lorentz spaces, defined with respect to the two-dimensional decreasing rearrangement, to be normable. To this end, it is also useful to study the mixed norm Lorentz spaces. Finally, we prove embeddings between all the classical, multidimensional, and mixed norm Lorentz spaces. Research partially supported by KAW 2000.0048 and STINT KU 2002-4025. Research partially supported by Grants MTM2004-02299, 2005SGR00556 and The Swedish Research Council no. 624-2003-571.     

Markov-type estimates for certain classes of constrained polynomials

Establishing Markov-type inequalities for the derivatives of polynomials with restricted zeros was initiated by P. Erdös [3] in 1940. Since then several authors proved similar estimates for the derivatives of polynomials of special type. In this paper we work on [–1, 1] and obtain Markov-type estimates for the derivatives of polynomials from a rather wide family of classes of constrained polynomials. In some cases the results turn out to be sharp.Communicated by George G. Lorentz.     

赋Orlicz范数的Orlicz—Lorentz空间的严格凸性

本文对于赋Ｌｕｘｅｍｂｕｒｇ范数的Ｏｒｌｉｃｚ－Ｌｏｒｅｎｔｚ空间     

Isotonicity of the Metric Projection by Lorentz Cone and Variational Inequalities

In this paper, we first discuss the geometric properties of the Lorentz cone and the extended Lorentz cone. The self-duality and orthogonality of the Lorentz cone are obtained in Hilbert spaces. These properties are fundamental for the isotonicity of the metric projection with respect to the order, induced by the Lorentz cone. According to the Lorentz cone, the quasi-sublattice and the extended Lorentz cone are defined. We also obtain the representation of the metric projection onto cones in Hilbert quasi-lattices. As an application, solutions of the classic variational inequality problem and the complementarity problem are found by the Picard iteration corresponding to the composition of the isotone metric projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. Our results generalize and improve various recent results obtained by many others.     

Markov-Bernstein type inequalities of multivariate polynomials with positive coefficients and applications

The Cauchy’s formula of entire functions f:C^k→C is used to establish Markov-Bernstein type inequalities of multivariate polynomials with positive coeffeicients on the k-dimensional simplex T_k⊂R^k and on the cube [0,1]^k. The main results generalize and improve those of G.G. Lorentz, etc. Some applications of these inequalities are also considered in polynomial constrained approximation.     

Convolution Inequalities in Lorentz Spaces

In this paper we study boundedness of the convolution operator in different Lorentz spaces. We obtain the limit case of the Young-O’Neil inequality in the classical Lorentz spaces. We also investigate the convolution operator in the weighted Lorentz spaces.