A Korovkin theorem in multivariate modular function spaces

Bardaro, Carlo; Mantellini, Ilaria

doi:10.1155/2009/863153

Cited by 37 publications

(70 citation statements)

References 21 publications

(30 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Also the case of not necessarily positive operators is considered, following an approach given in [5]. Our results extend Korovkin-type theorems given in [8,10,17,18] in the context of modular spaces and in [16] in the setting of ideal convergence. Note that at least the results concerning positive operators can be extended to more general kinds of convergence, not necessarily generated by free filters or regular matrix methods: among them we recall almost convergence [25].…”

Section: Introductionsupporting

confidence: 67%

See 1 more Smart Citation

Abstract Korovkin-type theorems in modular spaces and applications

et al. 2013

Self Cite

View full text Add to dashboard Cite

Abstract:We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the corresponding classical ones. MSC:40A35, 41A35, 46E30

show abstract

Section: Introductionsupporting

confidence: 67%

“…By applying the limit superior and taking into account property (ρ)-( * ), from (8) and (9) we obtain…”

Section: Theorem 42mentioning

confidence: 99%

Abstract Korovkin-type theorems in modular spaces and applications

et al. 2013

Self Cite

View full text Add to dashboard Cite

show abstract

“…Our main convergence theorems make use of the following result which is a consequence of a general Korovkin type theorem in abstract modular spaces proved in [5]. The proof of this theorem is mainly based on a modular density theorem of the subspace C(A) (see [14]).…”

Section: Let Us Consider the Functionsmentioning

confidence: 99%

“…In Sections 3 and 4, we obtain modular convergence theorems in the Orlicz spaces for the two operators. A key tool in order to obtain these convergence theorems is a multivariate version in modular spaces of the well known Korovkin theorem, proved recently in [5] (for the classical Korovkin approximation theory see e.g. [10] and [1]).…”

Section: Introductionmentioning

confidence: 99%

Multivariate moment type operators: approximation properties in Orlicz spaces

Bardaro¹,

Mantellini²

2008

Journal of Mathematical Inequalities

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper modular convergence theorems in Orlicz spaces for multivariate extensions of the one-dimensional moment operator are given and the order of modular convergence in modular Lipschitz classes is studied. IntroductionIn several papers (see [18] and some of its generalized versions were studied. In particular, these operators have nice pointwise approximation properties. Indeed they reduce the essential jump of the function f at a point s ∈ [0, 1] and they converge to f (s) when s is a Lebesgue point of f . Moreover the use of these operators in problems of calculus of variation was very wide. For example it is possible to show that the sequence M n f converges in variation or in lenght to f . Again the one-dimensional moment operator has interesting applications to the fractional calculus and it was used in the study of the fractional dimension of measurable sets (see [18], [12]). In [9] and [11] some bivariate versions of the above operator were studied in connection with the pointwise convergence and convergence with respect to some functional of calculus of variation as, for example, the surface area and perimeter of sets. Some of these extended operators have a kernel given by radial functions which characterizes them as "metric type operators". It is important to remark that in the earlier paper [17], some of the above properties were obtained in a general form by considering families of Urysohn type operators.Here we give some multidimensional versions of the moment type or metric type operators and we study their convergence properties in the general frame of the Orlicz spaces. This enables us to obtain corresponding results in L p -spaces. In particular weMathematics subject classification (2000): 41A35, 47G10, 46E30.

show abstract

“…Korovkin [15] established the sufficient conditions for the uniform convergence of (L n ) to a function f by using the test functions 1, x, x 2 . Many researchers have investigated these conditions for various operators defined on different spaces (see for instance [2,13,25]). Recently, Demirci and Orhan [9] introduced statistical relative uniform convergence of single sequences by using the notions of the natural density and the relative uniform convergence.…”

Section: Introductionmentioning

confidence: 99%