Integral-type operators on continuous function spaces on the real line

Altomare, Francesco; Milella, Sabina

doi:10.1016/j.jat.2007.11.002

Cited by 8 publications

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References 5 publications

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“…For additional properties of Gauss-Weierstrass operators, we refer, e.g., to [38] (see also a recent generalization given in [21][22][23]).…”

Section: Korovkin's Second Theorem and Something Elsementioning

confidence: 99%

Korovkin-type Theorems and Approximation by Positive Linear Operators

Altomare

2010

Preprint

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This survey paper contains a detailed self-contained introduction to Korovkin-type theorems and to some of their applications concerning the approximation of continuous functions as well as of L p -functions, by means of positive linear operators.The paper also contains several new results and applications. Moreover, the organization of the subject follows a simple and direct approach which quickly leads both to the main results of the theory and to some new ones.MSC: 41A36, 46E05, 47B65 Keywords: Korovkin-type theorem, positive operator, approximation by positive operators, Stone-Weierstrass theorem, (weighted) continuous function space, L p -space.8 Korovkin-type theorems in weighted continuous function spaces and in L p (X, µ) spaces 1359 Korovkin-type theorems and Stone-Weierstrass theorems 143 10 Korovkin-type theorems for positive projections 14611 Appendix: A short review of locally compact spaces and of some continuous function spaces on them 151References 1571 IntroductionKorovkin-type theorems furnish simple and useful tools for ascertaining whether a given sequence of positive linear operators, acting on some function space is an approximation process or, equivalently, converges strongly to the identity operator. Roughly speaking, these theorems exhibit a variety of test subsets of functions which guarantee that the approximation (or the convergence) property holds on the whole space provided it holds on them.The custom of calling these kinds of results "Korovkin-type theorems" refers to P. P. Korovkin who in 1953 discovered such a property for the functions 1, x and x 2 in the space C([0, 1]) of all continuous functions on the real interval [0, 1] as well as for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line ([77-78]).After this discovery, several mathematicians have undertaken the program of extending Korovkin's theorems in many ways and to several settings, including function spaces, abstract Banach lattices, Banach algebras, Banach spaces and so on. Such developments delineated a theory which is nowadays referred to as Korovkin-type approximation theory.This theory has fruitful connections with real analysis, functional analysis, harmonic analysis, measure theory and probability theory, summability theory and partial differential equations. But the foremost applications are concerned with constructive approximation theory which uses it as a valuable tool.Even today, the development of Korovkin-type approximation theory is far from complete, especially for those parts of it that concern limit operators different from the identity operator (see Problems 5.3 and 5.4 and the subsequent remarks).A quite comprehensive picture of what has been achieved in this field until 1994 is documented in the monographs of Altomare and Campiti ([8], see in particular Appendix D), Donner ([46]), Keimel and Roth ([76]), Lorentz, v. Golitschek and Makovoz ([83]). More recent results can be found, e.g., in [1], [9-15], [22], [47-52], [63], [71-74], [79], [114-116], [117...

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“…For additional properties of Gauss-Weierstrass operators, we refer, e.g., to [38] (see also a recent generalization given in [21][22][23]).…”

Section: Korovkin's Second Theorem and Something Elsementioning

confidence: 99%

Korovkin-type Theorems and Approximation by Positive Linear Operators

Altomare

2010

Preprint

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“…The above operators, which turn into the Gauss-Weierstrass operators when α = 1 and β = 0, have been introduced and studied in [4] in the onedimensional case. There we proved that the operators G n have interesting shape-preserving and regularity properties and, moreover, that the sequence (G n ) n≥1 is an approximation process on some continuous function spaces on the real line, with respect to weighted norms or to the norm of uniform convergence.…”

Section: Integral Operators Onmentioning

confidence: 99%

“…Recently in [4] and [5] we introduced and studied a generalization of Gauss-Weierstrass operators in the setting of weighted spaces of continuous functions on the real line.…”

Section: Introductionmentioning

confidence: 99%

On a sequence of integral operators on weighted L p spaces

Altomare

Milella

2008

Anal Math

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A b s t r a c t . Continuing some investigations started in previous papers, we introduce and study a sequence of multidimensional positive integral operators which generalize the Gauss-Weierstrass operators. We show that this sequence is an approximation process in some classes of weighted L p spaces on R N , N ≥ 1. Estimates of the rate of convergence are also obtained.Our mean tool is a Korovkin-type theorem which we establish in the context of L p (X, µ) spaces, X being a locally compact Hausdorff space and µ a regular positive Borel measure on X . Several examples are explicitly indicated as well.

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On Wachnicki’s Generalization of the Gauss–Weierstrass Integral

Abel

Agratini

2022

Trends in Mathematics

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Integral-type operators on continuous function spaces on the real line

Cited by 8 publications

References 5 publications

Korovkin-type Theorems and Approximation by Positive Linear Operators

Korovkin-type Theorems and Approximation by Positive Linear Operators

On a sequence of integral operators on weighted L p spaces

On Wachnicki’s Generalization of the Gauss–Weierstrass Integral

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