Convergence in variation for a homothetic modulus of smoothness in multidimensional setting (Q2910129)

In the paper under review, the author works with the concept of multidimensional \(\varphi\)-variation introduced by \textit{L.~Angeloni} and \textit{G.~Vinti} [J. Math. Anal. Appl. 349, No. 2, 317--334 (2009; Zbl 1154.26017)]. This \(\varphi\)-variation extends to the setting of Musielak-Orlicz \(\varphi\)-variation a multidimensional definition of variation introduced by \textit{C. Vinti} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 18, 201--231 (1964; Zbl 0129.10705)], based on Tonelli's approach. The author defines the space of functions of bounded \(\varphi\)-variation \({BV^{\varphi}}\) and the subspace \({AC^{\varphi}}\) of all the functions of bounded \(\varphi\)-variation which are locally \(\varphi\)-absolutely continuous.NEWLINENEWLINEThe \(\varphi\)-modulus of smoothness of a function \(f\) in \({BV^{\varphi}}\) is defined as NEWLINE\[NEWLINE\omega^{\varphi}(f,\delta):=\sup_{|\mathbf {1-t}|<\delta} V^{\varphi}[\tau_t f-f], NEWLINE\]NEWLINE NEWLINEwhere \(\delta>0\), \(V^{\varphi}\) denotes \(\varphi\)-variation, \((\tau_t)f(\mathbf{s})=f(\mathbf{st})\) for every \(\mathbf {st}\in \mathbb (R_0^+)^N\), \(\mathbf 1=(1,1,\dots, 1)\). This modulus of smoothness is a useful tool in investigations of the convergence in \(\varphi\)-variation for a class of Mellin-type operators. The main result of the paper (Theorem~5.1) states that, if \({f\in AC^{\varphi}}\), then there exists \(\lambda>0\) such that \(\lim_{\mathbf{|1-t|}\to 0^+}V^{\varphi}[\lambda(\tau_t f-f)]=0\), which implies that \(\lim_{\delta\to 0^+}\omega^{\varphi}(\lambda f,\delta)=0\).