Recent zbMATH articles in MSC 28A15 https://www.zbmath.org/atom/cc/28A15 2021-06-15T18:09:00+00:00 Werkzeug Theory of the $$(m,\sigma)$$-general functions over infinite-dimensional Banach spaces. https://www.zbmath.org/1460.46032 2021-06-15T18:09:00+00:00 "Asci, Claudio" https://www.zbmath.org/authors/?q=ai:asci.claudio Summary: In this paper, we introduce some functions, called $$(m,\sigma)$$-general, that generalize the $$(m,\sigma)$$-standard functions and are defined in the infinite-dimensional Banach space $$E_I$$ of the bounded real sequences $$\{x_n\}_{n\in I}$$, for some subset $$I$$ of $$\mathbb N$$. Moreover, we recall the main results about the differentiation theory over $$E_I$$, and we expose some properties of the $$(m,\sigma)$$-general functions. Finally, we study the linear $$(m,\sigma)$$-general functions, by introducing a theory that generalizes the standard theory of the $$m\times m$$ matrices. Change of variables' formula for the integration of the measurable real functions over infinite-dimensional Banach spaces. https://www.zbmath.org/1460.46033 2021-06-15T18:09:00+00:00 "Asci, Claudio" https://www.zbmath.org/authors/?q=ai:asci.claudio Summary: In this paper we study, for any subset $$I$$ of $$\mathbb N^*$$ and for any strictly positive integer $$k$$, the Banach space $$E_I$$ of the bounded real sequences $$\{x_n\}_{n\in I}$$, and a measure over $$(\mathbb R^I, \mathcal{B}^{(I)})$$ that generalizes the $$k$$-dimensional Lebesgue one. Moreover, we recall the main results about the differentiation theory over $$E_I$$. The main result of our paper is a change of variables formula for the integration of the measurable real functions on $$(\mathbb R^I, \mathcal{B}^{(I)})$$. This change of variables is defined by some functions over an open subset of $$E_J$$, with values on $$E_I$$, called $$(m,\sigma)$$-general, with properties that generalize the analogous ones of the finite-dimensional diffeomorphisms. Differentiating Orlicz spaces with rectangles having fixed shapes in a set of directions. https://www.zbmath.org/1460.42024 2021-06-15T18:09:00+00:00 "D'Aniello, Emma" https://www.zbmath.org/authors/?q=ai:daniello.emma "Moonens, Laurent" https://www.zbmath.org/authors/?q=ai:moonens.laurent Summary: In the present note, we examine the behavior of some homothecy-invariant differentiation basis of rectangles in the plane satisfying the following requirement: for a given rectangle to belong to the basis, the ratio of the largest of its side-lengths by the smallest one (which one calls its shape) has to be a fixed real number depending on the angle between its longest side and the horizontal line (yielding a shape-function). Depending on the allowed angles and the corresponding shape-function, a basis may differentiate various Orlicz spaces. We here give some examples of shape-functions so that the corresponding basis differentiates $$L \log (L(\mathbb{R}^2)$$, and show that in some model'' situations, a fast-growing shape function (whose speed of growth depends on $$\alpha > 0$$) does not allow the differentiation of $$L \log^{\alpha}L(\mathbb R^2)$$.