A Characterization of a Modulus of Smoothness in Multidimensional Setting

Angeloni, Laura

Angeloni, Laura

Bollettino dell'Unione Matematica Italiana, Tome 4 (2011), p. 79-108 / Harvested from Biblioteca Digitale Italiana di Matematica

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Résumé

A classical result of approximation theory states that $\lim_{\delta\to 0}\omega(f,\delta)=0$ , where $\omega$ is the modulus of smoothness of $f$ defined by means of the variation functional, if and only if $f$ is absolutely continuous. Such theorem is crucial in order to obtain results about convergence and order of approximation for linear and non-linear integral operators in BV-spaces. It was an open problem to extend the above result to the setting of $\varphi$ -variation in the multidimensional frame. In this paper, working with a concept of multidimensional W-variation introduced in [3], we prove that an analogous characterization holds for the multidimensional $\varphi$ -modulus of smoothness.

Publié le : 2011-02-01

MR 2797467 | Zbl 1237.26011

@article{BUMI_2011_9_4_1_79_0,
     author = {Laura Angeloni},
     title = {A Characterization of a Modulus of Smoothness in Multidimensional Setting},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {4},
     year = {2011},
     pages = {79-108},
     zbl = {1237.26011},
     mrnumber = {2797467},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2011_9_4_1_79_0}
}

Angeloni, Laura. A Characterization of a Modulus of Smoothness in Multidimensional Setting. Bollettino dell'Unione Matematica Italiana, Tome 4 (2011) pp. 79-108. http://gdmltest.u-ga.fr/item/BUMI_2011_9_4_1_79_0/

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