Angeloni, Laura:
A Characterization of a Modulus of Smoothness in Multidimensional Setting
Bollettino dell'Unione Matematica Italiana Serie 9 4 (2011), fasc. n.1, p. 79-108, (English)
pdf (418 Kb), djvu (268 Kb). | MR 2797467 | Zbl 1237.26011
Sunto
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.
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