Luxemburg Norm Research Articles

Orlicz spaces with $s$-norms that are abstract M-spaces

Let \Phi be an Orlicz function and L^{\Phi} (X, \Sigma, \mu) the corresponding Orlicz space on a non-atomic, \sigma -finite, complete measure space (X,\Sigma,\mu) . We describe those Orlicz spaces equipped with s -norms that are abstract M-spaces. Our study generalizes and unifies the results that have been obtained for the Orlicz norm, the Luxemburg norm, and the p -Amemiya norms separately.

Smooth, strongly smooth and very smooth points of Orlicz-Bochner sequence spaces endowed with the Luxemburg norm

In this paper, necessary and sufficient conditions for smooth, strongly smooth and very smooth points of Orlicz-Bochner sequence spaces endowed with the Luxemburg norm are given. Then criteria for smoothness, strong smoothness and very smoothness of Orlicz-Bochner sequence spaces are obtained as corollary.

Kadec–Klee property with respect to the local convergence in measure of Orlicz–Lorentz spaces

AbstractIn this paper, we find criteria for the Kadec–Klee property with respect to the local convergence in measure in both Orlicz–Lorentz spaces as well as their subspaces of order continuous elements. In the case of Orlicz norm, the presented results are new, whereas in the case of Luxemburg norm, we rely heavily on known results, which we show for the first time as a whole. Finally, we apply the obtained results to Orlicz spaces.

Homogeneous eigenvalue problems in Orlicz-Sobolev spaces

In this article we consider a homogeneous eigenvalue problem ruled by the fractional $g$-Laplacian operator whose Euler-Lagrange equation is obtained by minimization of a quotient involving Luxemburg norms. We prove existence of an infinite sequence of variational eigenvalues and study its behavior as the fractional parameter $s\uparrow 1$ among other stability results.

On the convexity coefficient of Musielak–Orlicz function spaces equipped with the Orlicz norm

AbstractIn Hudzik and Landes, the convexity coefficient of Musielak–Orlicz function spaces over a non‐atomic measure space equipped with the Luxemburg norm is computed whenever the Musielak–Orlicz functions are strictly convex see [6]. In this paper, we extend this result to the case of Musielak–Orlicz spaces equipped with the Orlicz norm. Also, a characterization of uniformly convex Musielak–Orlicz function spaces as well as k‐uniformly convex Musielak–Orlicz spaces equipped with the Orlicz norm is given.

Extreme points in Orlicz spaces equipped with s‐norms and closedness

AbstractLet Φ be an Orlicz function and be the corresponding Orlicz space on a non‐atomic, σ‐finite, complete measure space . We describe the extreme points of unit ball of Orlicz spaces equipped with the s‐norm. We also investigate the closedness of the set of extreme points of unit ball. Our study generalizes and unifies the results that have been obtained for the Orlicz norm, the Luxemburg norm and the p‐Amemiya norm separately. We further classify the outer functions that generate s‐norms with respect to a constant.

Daugavet and diameter two properties in Orlicz-Lorentz spaces

In this article, we study the diameter two properties (D2Ps), the diametral diameter two properties (diametral D2Ps), and the Daugavet property in Orlicz-Lorentz spaces equipped with the Luxemburg norm. First, we characterize the Radon-Nikodým property of Orlicz-Lorentz spaces in full generality by considering all finite real-valued Orlicz functions. To show this, the fundamental functions of their Köthe dual spaces defined by extended real-valued Orlicz functions are computed. We also show that if an Orlicz function does not satisfy the appropriate Δ2-condition, the Orlicz-Lorentz space and its order-continuous subspace have the strong diameter two property. Consequently, given that an Orlicz function is an N-function at infinity, the same condition characterizes the diameter two properties of Orlicz-Lorentz spaces as well as the octahedralities of their Köthe dual spaces. The Orlicz-Lorentz function spaces with the Daugavet property and the diametral D2Ps are isometrically isomorphic to L1 when the weight function is regular. In the process, we observe that every locally uniformly nonsquare point is not a Δ-point. This fact provides another class of real Banach spaces without Δ-points. As another application, it is shown that for Orlicz-Lorentz spaces equipped with the Luxemburg norm defined by an N-function at infinity, their Köthe dual spaces do not have the local diameter two property, and so as other (diametral) diameter two properties and the Daugavet property.

On the comparison of the Marcinkiewicz and Luxemburg norms of the exponential space

On the comparison of the Marcinkiewicz and Luxemburg norms of the exponential space

Wavelet approximation in Orlicz spaces

Approximation properties of wavelet decompositions in the Orlicz spaces are investigated. Both dual wavelet frames and frame-like wavelet systems are considered. The order of approximation (in the sense of modular convergence) of wavelet frame decompositions satisfying a number of natural conditions is found for an arbitrary Orlicz space. For the Orlicz spaces satisfying Δ2-condition, an error estimate providing approximation order of such wavelet expansions is given in terms of the Luxemburg norm. Similar results are obtained for appropriate frame-like systems under the assumption that an Orlicz space satisfies Δ′-condition.

An improvement of Bernstein’s inequality for functions in Orlicz spaces with smooth Fourier image

Let Φ:[0,+∞)→[0,+∞] be an arbitrary Young function and K be a compact set in ℝn having (O)-property. Then there exists a constant CK,Φ<∞ independent of f such that ∥Dαf∥(Φ)≤CK,Φsupz∈K|zα|∥f∥ℒK,3 for all α∈ℤ+n and f∈ℒK,3, where ℒK,3={f∈𝒮′(ℝn):suppf^⊂K,D(3,3,…,3)f^∈C(ℝn)}, ∥f∥ℒK,3=∥D(3,3,…,3)f^∥∞, f^ is the Fourier transform of f and ∥⋅∥(Φ) is the Luxemburg norm. As an application, a new Paley–Wiener theorem is given.

MODULAR BLOK DI RUANG BARISAN TERJUMLAH CESARO ORLICZ

On the Cesaro summable of orde-p sequence space, if the fuction is replaced by Orlicz function, it is not always easy to define norm in the space. In this paper, we study some properties of the Cesaro Orlicz summable sequence space. First, on the space we define a modular and its the luxemburg norm, and then some topological properties is explored. The results show that the sequence spaces is modular complete and nom complete. In addition, the space is a BK-space but not an AK-space.

On the comparison of the Lorentz and Luxemburg norms of the Lp log L space

On the comparison of the Lorentz and Luxemburg norms of the Lp log L space

Orthogonality in nonseparable rearrangement-invariant spaces

Let be a nonseparable rearrangement-invariant space and let be the closure of the space of bounded functions in . Elements of orthogonal to , that is, elements , , such that for each , are investigated. The set of orthogonal elements is characterized in the case when is a Marcinkiewicz or an Orlicz space. If an Orlicz space is considered with the Luxemburg norm, then the set is the algebraic sum of and . Each nonseparable rearrangement-invariant space such that is shown to contain an asymptotically isometric copy of the space . Bibliography: 17 titles.

Monotonicity properties and solvability of dominated best approximation problem in Orlicz spaces equipped with s-norms

In the paper, Wisła (J Math Anal Appl 483(2):123659, 2020, 10.1016/j.jmaa.2019.123659), it was proved that the classical Orlicz norm, Luxemburg norm and (introduced in 2009) p-Amemiya norm are, in fact, special cases of the s-norms defined by the formula left| xright| _{Phi ,s}=inf _{k>0}frac{1}{k}sleft( int _T Phi (kx)dmu right) , where s and Phi are an outer and Orlicz function respectively and x is a measurable real-valued function over a sigma -finite measure space (T,Sigma ,mu ). In this paper the strict monotonicity, lower and upper uniform monotonicity and uniform monotonicity of Orlicz spaces equipped with the s-norm are studied. Criteria for these properties are given. In particular, it is proved that all of these monotonicity properties (except strict monotonicity) are equivalent, provided the outer function s is strictly increasing or the measure mu is atomless. Finally, some applications of the obtained results to the best dominated approximation problems are presented.

Extreme points of the Besicovitch--Orlicz space of almost periodic functions equipped with the Luxemburg norm

We investigate which points in the unit sphere of the Besicovitch--Orlicz space of almost periodic functions, equipped with the Luxemburg norm, are extreme points. Sufficient conditions for the strict convexity of this space are also given.

Smoothness of Orlicz function spaces equipped with the p-Amemiya norm

In this paper, we will use the convex modular rho ^{*}(f) to investigate Vert fVert _{Psi ,q}^{*} on (L_{Phi })^{*} defined by the formula Vert fVert _{Psi ,q}^{*}=inf _{k>0}frac{1}{k}s_{q}(rho ^{*}(kf)), which is the norm formula in Orlicz dual spaces equipped with p-Amemiya norm. The attainable points of dual norm Vert fVert _{Psi ,q}^{*} are discussed, the interval for dual norm Vert fVert _{Psi ,q}^{*} attainability is described. By presenting the explicit form of supporting functional, we get sufficient and necessary conditions for smooth points. As a result, criteria for smoothness of L_{Phi ,p}~(1le ple infty ) is also obtained. The obtained results unify, complete and extended as well the results presented by a number of paper devoted to studying the smoothness of Orlicz spaces endowed with the Luxemburg norm and the Orlicz norm separately.

Strongly Proximinal Subspaces in Orlicz Function Space

In this paper, we prove that if Y is a separable proximinal subspace of , then Yis strongly proximinal in if and only if L(U,Y) is strongly L(U,Y) proximinal in. where L(U,Y) is an Orlicz function space with Luxemburg norm.

Fixed point properties and reflexivity in variable Lebesgue spaces

In this paper the weak fixed point property (w-FPP) and the fixed point property (FPP) in Variable Lebesgue Spaces are studied. Given (Ω,Σ,μ) a σ-finite measure and p(⋅) a variable exponent function, the w-FPP is completely characterized for the variable Lebesgue space Lp(⋅)(Ω) in terms of the exponent function p(⋅) and the absence of an isometric copy of L1[0,1]. In particular, every reflexive Lp(⋅)(Ω) has the FPP and our results bring to light the existence of some nonreflexive variable Lebesgue spaces satisfying the w-FPP, in sharp contrast with the classic Lebesgue Lp-spaces. In connection with the FPP, we prove that Maurey's result for L1-spaces can be extended to the larger class of variable Lp(⋅)(Ω) spaces with order continuous norm, that is, every reflexive subspace of Lp(⋅)(Ω) has the FPP. Nevertheless, Maurey's converse does not longer hold in the variable setting, since some nonreflexive subspaces of Lp(⋅)(Ω) satisfying the FPP can be found. As a consequence, we discover that several nonreflexive Nakano sequence spaces ℓpn do have the FPP endowed with the Luxemburg norm. As far as the authors are concerned, this family of sequence spaces gives rise to the first known nonreflexive classic Banach spaces enjoying the FPP without requiring of any renorming procedure. The failure of asymptotically isometric copies of ℓ1 in Lp(⋅)(Ω) is also analyzed.

Bounds on tail probabilities for quadratic forms in dependent sub-gaussian random variables

We show bounds on tail probabilities for quadratic forms in sub-gaussian non-necessarily independent random variables. Our main tool will be estimates of the Luxemburg norms of such forms. This will allow us to formulate the above-mentioned bounds. As an example we give estimates of the excess loss in fixed design linear regression in dependent observations.

When and where the Orlicz and Luxemburg (quasi-) norms are equivalent?

We study the equivalence between the Orlicz and Luxemburg (quasi-) norms in the context of the generalized Orlicz spaces associated to an N-function Φ and a (quasi-) Banach function space X over a positive finite measure μ. We show that the Orlicz and the Luxemburg spaces do not coincide in general, and also that under mild requirements (σ-Fatou property, strictly monotone renorming) the coincidence holds. We use as a technical tool the classes LwΦ(m), LΦ(m) and LΦ(‖m‖) of Orlicz spaces of scalar integrable functions with respect to a Banach-space-valued countably additive vector measure m, providing also some new results on these spaces.