scholarly journals A General Approximation Approach for the Simultaneous Treatment of Integral and Discrete Operators

2018 ◽  
Vol 18 (4) ◽  
pp. 705-724 ◽  
Author(s):  
Gianluca Vinti ◽  
Luca Zampogni
Keyword(s):  
Orlicz Spaces ◽  
General Setting ◽  
Integral Operators ◽  
Topological Groups ◽  
Simultaneous Treatment ◽  
Modular Convergence ◽  
Unitary Approach ◽  
Discrete Operators ◽  
Simultaneous Study ◽  
Continuous Functionals

AbstractIn this paper, we give a unitary approach for the simultaneous study of the convergence of discrete and integral operators described by means of a family of linear continuous functionals acting on functions defined on locally compact Hausdorff topological groups. The general family of operators introduced and studied includes very well-known operators in the literature. We give results of uniform convergence and modular convergence in the general setting of Orlicz spaces. The latter result allows us to cover many other settings as the {L^{p}}-spaces, the interpolation spaces, the exponential spaces and many others.

scholarly journals Approximation Results for a General Class of Kantorovich Type Operators

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Gianluca Vinti ◽  
Luca Zampogni
Keyword(s):  
Orlicz Spaces ◽  
Integral Operators ◽  
Topological Groups ◽  
Locally Compact ◽  
Simultaneous Approach ◽  
Modular Convergence ◽  
Convergence Results ◽  
Discrete Operators ◽  
General Convergence ◽  
Locally Compact Topological Groups

AbstractWe introduce and study a family of integral operators in the Kantorovich sense acting on functions defined on locally compact topological groups. We obtain convergence results for the above operators with respect to the pointwise and uniform convergence and in the setting of Orlicz spaces with respect to the modular convergence. Moreover, we show how our theory applies to several classes of integral and discrete operators, as sampling, convolution and Mellin type operators in the Kantorovich sense, thus obtaining a simultaneous approach for discrete and integral operators. Further, we obtain general convergence results in particular cases of Orlicz spaces, as L

Convergence of sampling Kantorovich operators in modular spaces with applications

2020 ◽  
Author(s):  
Danilo Costarelli ◽  
Gianluca Vinti
Keyword(s):  
Orlicz Spaces ◽  
General Setting ◽  
Interpolation Spaces ◽  
Convergence Theorems ◽  
Modular Spaces ◽  
Modular Convergence ◽  
Kantorovich Operators

Abstract In the present paper we study the so-called sampling Kantorovich operators in the very general setting of modular spaces. Here, modular convergence theorems are proved under suitable assumptions, together with a modular inequality for the above operators. Further, we study applications of such approximation results in several concrete cases, such as Musielak–Orlicz and Orlicz spaces. As a consequence of these results we obtain convergence theorems in the classical and weighted versions of the $$L^p$$ L p and Zygmund (or interpolation) spaces. At the end of the paper examples of kernels for the above operators are presented.

scholarly journals Quantitative Estimates for Nonlinear Sampling Kantorovich Operators

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Nursel Çetin ◽  
Danilo Costarelli ◽  
Gianluca Vinti
Keyword(s):  
Orlicz Spaces ◽  
General Setting ◽  
Modulus Of Smoothness ◽  
Direct Approach ◽  
Order Of Approximation ◽  
Lipschitz Classes ◽  
General Frame ◽  
Quantitative Estimates ◽  
Nonlinear Sampling ◽  
Kantorovich Operators

AbstractIn this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of smoothness in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in $$L^{p}$$ L p -spaces, $$1\le p<\infty $$ 1 ≤ p < ∞ , and in other well-known instances of Orlicz spaces, such as the Zygmung and the exponential spaces. Further, the qualitative order of approximation has been obtained assuming f in suitable Lipschitz classes. The above estimates achieved in the general setting of Orlicz spaces, have been also improved in the $$L^p$$ L p -case, using a direct approach suitable to this context. At the end, we consider the particular cases of the nonlinear sampling Kantorovich operators constructed by using some special kernels.

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

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.

scholarly journals Remarks on a nonlinear nonlocal operator in Orlicz spaces

2019 ◽  
Vol 9 (1) ◽  
pp. 305-326 ◽  
Author(s):  
Ernesto Correa ◽  
Arturo de Pablo
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Sobolev Inequality ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Laplacian Operator ◽  
Nonlocal Operator ◽  
Generalized Eigenvalue

Abstract We study integral operators $\mathcal{L}u\left( \chi \right)=\int{_{_{\mathbb{R}}\mathbb{N}}\psi \left( u\left( x \right)-u\left( y \right) \right)J\left( x-y \right)dy}$of the type of the fractional p-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincaré inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel J considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem $\mathcal{L}u=f$in a bounded domain $\Omega ,$and boundary condition u ≡ 0 on ${{\Omega }^{c}};$both cases f = f(x) and f = f(u) are considred, including the generalized eigenvalue problem $f\left( u \right)=\lambda \psi \left( u \right).$

scholarly journals A Korovkin theorem in multivariate modular function spaces

2009 ◽  
Vol 7 (2) ◽  
pp. 105-120 ◽  
Author(s):  
Carlo Bardaro ◽  
Ilaria Mantellini
Keyword(s):  
Orlicz Spaces ◽  
Integral Operators ◽  
Modular Function ◽  
Function Spaces ◽  
Korovkin Theorem ◽  
Modular Function Spaces

In this paper a modular version of the classical Korovkin theorem in multivariate modular function spaces is obtained and applications to some multivariate discrete and integral operators, acting in Orlicz spaces, are given.

scholarly journals Homogeneous operators and homogeneous integral operators

2021 ◽  
Author(s):  
Zhirayr Avetisyan ◽  
Alexey Karapetyants
Keyword(s):  
Integral Operator ◽  
Metric Spaces ◽  
General Setting ◽  
Integral Operators ◽  
Homogeneous Operator ◽  
Homogeneous Operators

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure - measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples - classes of homogeneous integral operators on various metric spaces, and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article - various specific cases - illustrate general statements and results given in the paper and at the same time are of interest in their own way.

scholarly journals Estimates of multilinear singular integral operators and mean oscillation

2014 ◽  
Vol 95 (109) ◽  
pp. 201-214
Author(s):  
Lanzhe Liu
Keyword(s):  
Integral Operator ◽  
Singular Integral Operator ◽  
Singular Integral ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Multilinear Operators ◽  
Lebesgue Spaces ◽  
Marcinkiewicz Operator ◽  
Mean Oscillation

We prove the boundedness properties for some multilinear operators related to certain integral operators from Lebesgue spaces to Orlicz spaces. The operators include Calder?n-Zygmund singular integral operator, Littlewood-Paley operator and Marcinkiewicz operator.

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