On a sequence of integral operators on weighted L p spaces

Altomare, Francesco; Milella, Sabina

doi:10.1007/s10476-008-0401-5

Cited by 3 publications

(8 citation statements)

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“…In this section, we shall state an approximation formula for the semigroup considered in Theorem 3.1, by means of iterates of some integral operators that were recently introduced in [9] and that generalized the classical Gauss-Weierstrass operators.…”

Section: Approximation By Integral-type Operatorsmentioning

confidence: 99%

“…The operators (G n ) n≥1 have been introduced in [9]. There it was proved that they are an approximation process in suitable L p spaces.…”

Section: Approximation By Integral-type Operatorsmentioning

confidence: 99%

“…. , pr N }, then G n (h) = h for every n ≥ 1 (see [9,Lemma 3.4], and hence T m (t)(h) = h for every t ≥ 0, i.e., each T m (t) maps affine functions into affine functions.…”

Section: Some Properties Of the Semigroup And The Associated Markov Pmentioning

confidence: 99%

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Multiplicative perturbations of the Laplacian and related approximation problems

2011

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Of concern are multiplicative perturbations of the Laplacian acting on weighted spaces of continuous functions on R N , N ≥ 1. It is proved that such differential operators, defined on their maximal domains, are pre-generators of positive quasicontractive C 0 -semigroups of operators that fulfill the Feller property. Accordingly, these semigroups are associated with a suitable probability transition function and hence with a Markov process on R N . An approximation formula for these semigroups is also stated in terms of iterates of integral operators that generalize the classical Gauss-Weierstrass operators. Some applications of such approximation formula are finally shown concerning both the semigroups and the associated Markov processes.

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Section: Approximation By Integral-type Operatorsmentioning

confidence: 99%

“…The operators (G n ) n≥1 have been introduced in [9]. There it was proved that they are an approximation process in suitable L p spaces.…”

Section: Approximation By Integral-type Operatorsmentioning

confidence: 99%

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Multiplicative perturbations of the Laplacian and related approximation problems

2011

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“…B Sabina Milella sabina.milella@uniba.it 1 Dipartimento di Matematica, Università degli Studi di Bari "A. Moro", Via E. Orabona, 4, 70125 Bari, Italy A first multidimensional version of the G n s has been introduced in [8], in the setting of weighted L p spaces and weighted spaces of continuous functions on R d . These operators have been used in [9] to approximate and investigate Feller semigroups generated by bounded multiplicative perturbations of the Laplacian.…”

Section: Introductionmentioning

confidence: 99%

“…The coefficients a i j , b i and c are arbitrary real continuous functions on R d , thus including the results of [6,8,9].…”

Section: Introductionmentioning

confidence: 99%

Multivariate approximation by modified Gauss-Weierstrass operators

Milella

2015

Rend. Circ. Mat. Palermo

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We consider a generalization (G n ) n of the classical sequence of Gauss-Weierstrass operators, in order to obtain a Voronovskaja-type formula involving the general differentialThese operators have nice approximation, regularity and shape-preserving properties. In particular, by means of G n and its powers, we characterize the domains of all the realizations of A generating a positive C 0 -semigroup and we approximate such semigroup. We also discuss some qualitative properties of the semigroup as positivity, convexity and regularity preservation.

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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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On a sequence of integral operators on weighted L p spaces

Cited by 3 publications

References 6 publications

Multiplicative perturbations of the Laplacian and related approximation problems

Multiplicative perturbations of the Laplacian and related approximation problems

Multivariate approximation by modified Gauss-Weierstrass operators

Korovkin-type Theorems and Approximation by Positive Linear Operators

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