Luxemburg Norm Research Articles

UGD property of Musielak-Orlicz sequence spaces

By the properties of the Musielak-Orlicz function's sequence, the necessary and sufficient condition for uniform Gateaux differential (UGD) property of Musielak-Orlicz sequence spaces equipped with the Luxemburg norm and a criterion for weakly uniform rotundity of Musielak-Orlicz sequence space with Orlicz norm are given.

Jung constants of Orlicz sequence spaces

Estimation of the Jung constants of Orlicz sequence spaces equipped with either the Luxemburg norm or the Orlicz norm is given. The exact values of the Jung constants of a class of reflexive Orlicz sequence spaces are found by using new quantitative indic

On k‐nearly uniform convex property in generalized Cesàro sequence spaces

We define a generalized Cesàro sequence space ces(p), where p = (pk) is a bounded sequence of positive real numbers, and consider it equipped with the Luxemburg norm. The main purpose of this paper is to show that ces(p) is k‐nearly uniform convex (k‐NUC) for k ≥ 2 when limn→∞infpn > 1. Moreover, we also obtain that the Cesàro sequence space cesp(where 1 < p < ∞) is k‐NUC, kR, NUC, and has a drop property.

ON SOME EXPRESSIONS FOR THE RIESZ ANGLE OF ORLICZ SEQUENCE SPACES

We obtain some expressions for the Riesz angles of Orlicz sequence spaces endowed with the Luxemburg and Orlicz norms and show some practical ways of computation.

Norm Attaining Operators on Some Classical Banach Spaces

We show that for the Köthe space X = c0 + 𝓁1(w), equipped with the Luxemburg norm, the set of norm attaining operators from X into any infinite-dimensional strictly convex Banach space Y is not dense in the space of all bounded operators. The same assertion holds for any infinite-dimensional L1(μ). This gives the first example of a classical space X satisfying the previous property. We also prove that all the spaces c0 + 𝓁1(w) are isomorphic for a large class of weights w.

RATIONAL APPROXIMATION IN L φ SPACES ON A FINITE UNION OF DISJOINT INTERVALS

ABSTRACT In this paper we study the convergence of the best rational approximants, to a function f, respect to the Luxemburg norm on a finite union of disjoint intervals, with diameters shrinking to zero. This work extends previous results on Lp spaces. We also prove existence of best local quasi-rational approximants on a finite point set.

On the nonsquare constants of L(Φ)[0,+1)

Let L(Φ)[0,+1) be the Orlicz function space generated by N−function Φ(u) with Luxemburg norm. We show the exact nonsquare constant of it when the right derivative φ(t) of Φ(u) is convex or concave.

Averaging operators on l^{p_n} and L^p(x)

We consider the generalized Lebesgue space Lp(x) and its discrete analogue l{pn} , each given the appropriate Luxemburg norm. Let Tk be the averaging operator given by (Tka)n = 1 k (an + an+1 + · · · + an+k−1), a = {an} ∈ l{pn}. We show that the Tk are uniformly bounded from l {pn} into l{pn} under certain assumptions on pn and find a counter–example to show that Tk need not be bounded if these assumptions are not satisfied. Moreover, we construct a bounded Lipschitz function p(x) on [0,∞) such that the operator Ts given, for each Tsf (x) = 1 s ∫ s 0 f (t)dt is unbounded on Lp(x) for all s > 0 . Mathematics subject classification (2000): 46E30,26D15.

On the Duality Mapping Sets in Orlicz Sequence Spaces

The criteria for the weak compactness of duality mapping sets J(x) = {f∈X* : = ∥f∥2 = ∥x∥2} in Orlicz sequence spaces endowed either with the Luxemburg norm or with the Orlicz norm are obtained.

Weak estimates for commutators of fractional integral operators

By introducing a kind of maximal operator of the fractional order associated with the mean Luxemburg norm and using the technique of the sharp function, the weak type L log L estimates for the commutators of the fractional integral operator and the related maximal operator are established.

Smooth, very smooth and strongly smooth points in Musielak-Orlicz sequence spaces

Criteria for smooth points, very smooth points and strongly smooth points in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm are given.

Some results on packing in Orlicz sequence spaces

We present monotonicity theorems for index functions of N-fuctions, and obtain formulas for exact values of packing constants. In particular, we show that the Orlicz sequence space l (N) generated by the N-function N(v) = (1 +jvj) ln(1 +jvj) jvj with Luxemburg norm has the Kottman constant K(l (N) ) = N 1 (1)=N 1 (1=2), which answers M. M. Rao and Z. D. Ren's (8) problem.

On CLUR points of Orlicz spaces

CriteriaforcompactlylocallyuniformlyrotundpointsinOrliczspacesare given. Let (X, kk ) be a Banach space. S(X) and B(X) are the unit sphere and unit ball of X respectively. A point x on S(X) is said to be a locally uniformly rotund point (LUR point) provided that {xn} ⊂ X, k xnk → 1 and k xn + xk → 2 imply k xn − xk → 0. Recently Y. A. Cui, H. Hudzik and C. Meng (2) introduced the concept of compactly locally uniformly rotund point (CLUR point for short). x ∈ S(X) is said to be a CLUR point provided that {xn} ⊂ X, k xnk → 1 and k xn + xk → 2 imply that {xn} is a compact set. Obviously, if every point on S(X) is LUR (CLUR) then X is a LUR (CLUR) space. In 1975, B. B. Panda and O. P. Kapoor (4) proved that CLUR implies the Kadec-Klee property and X is LUR iff X is CLUR and strictly rotund. In 1984, J. Y. Fu and W. Y. Zhang (3) showed that if X ∗∗ is CLUR and K is a non-null closed, convex Chebyshev set in X then the projection on K is continuous. Y. A. Cui et al. (2) obtained a criterion for a point to be CLUR in Orlicz sequence spaces equipped with the Luxemburg norm. In this paper we will discuss the CLUR points in Orlicz sequence spaces equipped with the Orlicz norm and in Orlicz function spaces equipped with both the Luxemburg and Orlicz norm. A mapping M : (−∞, ∞) → (0, ∞) is said to be an N-function if it is even, convex, vanishing only at zero and such that limu→0 M(u)/u = 0 and limu→∞ M(u)/u = ∞. An interval (a, b) is called a structurally affine interval (SAI for short) of M provided that M is affine on ( a, b) and it is not affine on ( a − s, b) or (a, b + s) for any s > 0. For an N-function M we

Contractive projections in Orlicz sequence spaces

We characterize norm‐one complemented subspaces of Orlicz sequence spaces ℓM equipped with either Luxemburg or Orlicz norm, provided that the Orlicz function M is sufficiently smooth and sufficiently different from the square function. We measure smoothness of M using AC1 and AC2 classes introduced by Maleev and Troyanski in 1991, and the condition for M to be different from a square function is essentially a requirement that the second derivative M″ of M cannot have a finite nonzero limit at zero. This paper treats the real case; the complex case follows from previously known results.

Weak orthogonality and weak property (β) in some banach sequence spaces

It is proved that a Köthe sequence space is weakly orthogonal if and only if it is order continuous. Criteria for weak property (β) in Orlicz sequence spaces in the case of the Luxemburg norm as well as the Orlicz norm are given.

Monotone coefficients and monotonicity of Orlicz spaces

The criteria for uniform monotonicity, locally uniformly monotonicity and monotonicity of of Orlicz spaces with Luxemburg and Orlicz norms are given. The monotone coefficients of a point and of the spaces are computed.

On Some Uniform Convexities and Smoothness in Certain Sequence Spaces

It is proved that any Banach space X with property A^e_2 has property A_2 and that a Banach space X is nearly uniformly smooth if and only if it is nearly uniformly *- smooth and weakly sequentially complete. It is shown that if X is a Köthe sequence space the dual of which contains no isomorphic copy of l_1 and has property A^e_2 , then X has the uniform Kadec-Klee property. Criteria for nearly uniformly convexity of Musielak-Orlicz spaces equipped with the Orlicz norm are presented. It is also proved that both properties nearly uniformly smoothness and nearly uniformly convexity for Musielak-Orlicz spaces equipped with the Luxemburg norm coincide with reflexivity. Finally, an interpretation of those results for Nakano spaces l^{(p_i)} (1 < p_i < \infty) is given.

Monotonicity Properties of Musielak–Orlicz Spaces and Dominated Best Approximation in Banach Lattices

Criteria for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and uniform monotonicity of a Musielak–Orlicz space endowed with the Amemiya norm and its subspace of order continuous elements are given in the cases of nonatomic and the counting measure space. To complete the results of Kurc (J. Approx. Theory69(1992), 173–187), criteria for upper local uniform monotonicity of these spaces equipped with the Luxemburg norm are also given. Some applications to dominated best approximation are presented.

Best monotoneL φ-approximations in several variables

Given a continuous function f defined on the unit cube ofR n and a convex function φt, φt(0)=0, φt(x)>0, for x>0, we prove that the set of best Lφt-approximations by monotone functions has exactly one element ft, which is also a continuous function. Moreover if the family of convex functions {φt}t>0 converges uniformly on compact sets to a function φ0, then the best approximation ft→f0 uniformly, as t→0, where f0 is the best approximation of f within the Orlicz space L0. The best approximations {ft} are obtained as well as minimizing integrals or the Luxemburg norm.

One-Complemented Subspaces of Real Sequence Spaces

Characterizations are given for 1-complemented hyperplanes of strictly monotone real Lorentz spaces and 1-complemented finite codimensional subspaces (which contain at least one basis element) of real Orlicz spaces equipped with either Luxemburg or Orlicz norm.