Approximation in Weighted Orlicz Spaces with a generating young function that might be non-convex

Let H0 be a selfadjoint operator such that Tr e−βH0 is of trace class for some β < 1, and let χɛ denote the set of ɛ-bounded forms, i.e., ∥(H0+C)−1/2−ɛX(H0+C)−1/2+ɛ∥ < C for some C > 0. Let χ := Span ∪ɛ∈(0,1/2]χɛ. Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρx = e−H0−X−ψx, X ∈ χ. We show that if Tr e−H0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] at ρ0; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ0, consisting of p-nearby states (those [Formula: see text] obeying C−1ρ1+p ≤ σ ≤ Cρ1 − p for some C > 1) admits a flat affine connection known as the (−1) connection, and the span of this set is part of the cotangent space of [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.

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Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces

Carpathian Mathematical Publications ◽

10.15330/cmp.13.2.326-339 ◽

2021 ◽

Vol 13 (2) ◽

pp. 326-339

Author(s):

H.H. Bang ◽

V.N. Huy

Keyword(s):

Fourier Transform ◽

Orlicz Space ◽

Differential Operators ◽

Orlicz Spaces ◽

Integral Operators ◽

Young Function ◽

The Fourier Transform

In this paper, we investigate the behavior of the sequence of $L^\Phi$-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function $f$ is called its spectrum and denoted by sp$(f)$). With $Q$ being a polynomial, we introduce the notion of $Q$-primitives, which will return to the notion of primitives if ${Q}(x)= x$, and study the behavior of the sequence of norm of $Q$-primitives of functions in Orlicz space $L^\Phi(\mathbb R^n)$. We have the following main result: let $\Phi $ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^\infty \subset L^\Phi(\mathbb R^n)$ satisfies $\mathcal{Q}^0f=f, {Q}(D)\mathcal{Q}^{m+1}f=\mathcal{Q}^mf$ for $m\in\mathbb{Z}_+$. Assume that sp$(f)$ is compact and $sp(\mathcal{Q}^{m}f)= sp(f)$ for all $m\in \mathbb{Z}_+.$ Then $$ \lim\limits_{m\to \infty } \|\mathcal{Q}^m f\|_{\Phi}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ {Q}({\bf x}) \bigl|. $$ The corresponding results for functions generated by differential operators and integral operators are also given.

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On Approximation by Linear Combinations of Modified Summation Operators of Integral Type in Orlicz Spaces

Mathematics ◽

10.3390/math7010006 ◽

2018 ◽

Vol 7 (1) ◽

pp. 6 ◽

Cited By ~ 1

Author(s):

Ling-Xiong Han ◽

Feng Qi

Keyword(s):

Orlicz Spaces ◽

Modulus Of Smoothness ◽

Young Function ◽

Integral Type ◽

Linear Combinations ◽

Equivalent Theorem

In this paper, the authors introduce the Orlicz spaces corresponding to the Young function and, by virtue of the equivalent theorem between the modified K-functional and modulus of smoothness, establish the direct, inverse, and equivalent theorems for linear combinations of modified summation operators of integral type in the Orlicz spaces.

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