A note on Lorentz–Zygmund spaces

2011 ◽  
Vol 18 (3) ◽  
pp. 533-548
Author(s):  
Hiroto Oba ◽  
Enji Sato ◽  
Yuichi Sato
Keyword(s):  
Real Line ◽  
Invariant Operator ◽  
Lorentz Spaces ◽  
Riesz Theorem ◽  
Translation Invariant ◽  
Bounded Linear ◽  
The Real ◽  
Translation Invariant Operator ◽  
Zygmund Spaces

Abstract Let 1 ≤ q < p < ∞, and ℝ be the real line. Hörmander showed that any bounded linear translation invariant operator from Lp (ℝ) to Lq (ℝ) is trivial. Blozinski obtained an analogy to Hörmander in Lorentz spaces on the real line. In this paper, we generalize Blozinski's result in Lorentz–Zygmund spaces. Also, Bochkarev proved an inequality related to the Hausdorff–Young–Riesz theorem in Lorentz spaces, and the sharpness of the inequality. We improve Bochkarev's inequality in Lorentz–Zygmund spaces, and prove the sharpness of our inequality.

scholarly journals Sums and products of intervals in ordered semigroups

2021 ◽  
Vol 29 (2) ◽  
pp. 187-198
Author(s):  
T. Glavosits ◽  
Zs. Karácsony
Keyword(s):  
Real Line ◽  
Ordered Semigroup ◽  
Translation Invariant ◽  
Famous Theorem ◽  
Ordered Semigroups ◽  
The Real ◽  
Ordered Ring

Abstract We show a simple example for ordered semigroup 𝕊 = 𝕊 (+,⩽) that 𝕊 ⊆ℝ (ℝ denotes the real line) and ]a, b[ + ]c, d[ = ]a + c, b + d[ for all a, b, c, d ∈ 𝕊 such that a < b and c < d, but the intervals are no translation invariant, that is, the equation c +]a, b[ = ]c + a, c + b[ is not always fulfilled for all elements a, b, c ∈ 𝕊 such that a < b. The multiplicative version of the above example is shown too. The product of open intervals in the ordered ring of all integers (denoted by ℤ) is also investigated. Let Ix := {1, 2, . . ., x} for all x ∈ ℤ+ and defined the function g : ℤ+ → ℤ+ by g ( x ) : = max { y ∈ ℤ + | I y ⊆ I x ⋅ I x } g\left( x \right): = \max \left\{ {y \in {\mathbb{Z}_ + }|{I_y} \subseteq {I_x} \cdot {I_x}} \right\} for all x ∈ ℤ+. We give the function g implicitly using the famous Theorem of Chebishev. Finally, we formulate some questions concerning the above topics.

Approximately dual and perturbation results for generalized translation-invariant frames on LCA groups

2018 ◽  
Vol 16 (03) ◽  
pp. 1850017 ◽  
Author(s):  
Yavar Khedmati ◽  
Mads S. Jakobsen
Keyword(s):  
Real Line ◽  
Locally Compact ◽  
Translation Invariant ◽  
The Real ◽  
Generalized Translation ◽  
Generalized Shift ◽  
Lca Groups

The results in this paper can be divided into three parts. First, we generalize the recent results of Benavente, Christensen and Zakowicz on approximately dual generalized shift-invariant frames on the real line to generalized translation-invariant (GTI) systems on locally compact abelian (LCA) groups. Second, we explain in detail how GTI frames can be realized as [Formula: see text]-frames. Finally, the known results on perturbation of [Formula: see text]-frames and results on perturbation of generalized shift-invariant systems are applied and extended to GTI systems on LCA groups.

Measurability properties of certain paradoxical subsets of the real line

2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Mariam Beriashvili
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Real Line ◽  
Translation Invariant ◽  
The Real

AbstractThe paper deals with the measurability properties of some classical subsets of the real line ℝ having an extra-ordinary descriptive structure: Vitali sets, Bernstein sets, Hamel bases, Luzin sets and Sierpiński sets. In particular, it is shown that there exists a translation invariant measure μ on ℝ extending the Lebesgue measure and such that all Sierpiński sets are measurable with respect to μ.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

AN OSCILLATION FUNCTION ON THE REAL LINE

2000 ◽  
Vol 26 (1) ◽  
pp. 237
Author(s):  
Duszyński
Keyword(s):  
Real Line ◽  
The Real

DERIVATION BASES ON THE REAL LINE (I)

1982 ◽  
Vol 8 (1) ◽  
pp. 67 ◽  
Author(s):  
Thomson
Keyword(s):  
Real Line ◽  
The Real

On finite sums of periodic functions

2020 ◽  
Vol 27 (2) ◽  
pp. 265-269
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Analytic Function ◽  
Real Line ◽  
Real Analytic Function ◽  
Periodic Functions ◽  
Uniform Limit ◽  
The Real ◽  
Pointwise Limit ◽  
Finite Sums ◽  
Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

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