In this paper we prove a convergence result for the modulus of smoothness in the frame of $BV^{\varphi}(\R^N_+)$, namely the space of functions of bounded $\varphi-$variation on $\R^N_+$ equipped with the logarithmic measure. In particular we prove that, similarly to what happens in the classical case of the Jordan variation, $\lim_{\delta \to 0^+} \omega^{\varphi}(\lambda f, \delta)=0,$ for some $\lambda >0$, if $f$ is $\varphi-$absolutely continuous. Here $\omega^{\varphi}(\lambda f, \delta):= \sup_{|{\tt 1-t}|\le\delta}V^{\varphi}[\lambda(\tau_{\tt t} f-f)]$, where $\tau_{\tt t}f({\tt s}):=f({\tt st})$, ${\tt s,t}\in\R^N_+$, is the dilation operator and $V^{\varphi}$ denotes a new concept of multivariate $\varphi-$variation in the sense of Tonelli.
A sufficient condition for the convergence of a certain modulus of smoothness in multidimensional setting
ANGELONI, Laura;VINTI, Gianluca
2013
Abstract
In this paper we prove a convergence result for the modulus of smoothness in the frame of $BV^{\varphi}(\R^N_+)$, namely the space of functions of bounded $\varphi-$variation on $\R^N_+$ equipped with the logarithmic measure. In particular we prove that, similarly to what happens in the classical case of the Jordan variation, $\lim_{\delta \to 0^+} \omega^{\varphi}(\lambda f, \delta)=0,$ for some $\lambda >0$, if $f$ is $\varphi-$absolutely continuous. Here $\omega^{\varphi}(\lambda f, \delta):= \sup_{|{\tt 1-t}|\le\delta}V^{\varphi}[\lambda(\tau_{\tt t} f-f)]$, where $\tau_{\tt t}f({\tt s}):=f({\tt st})$, ${\tt s,t}\in\R^N_+$, is the dilation operator and $V^{\varphi}$ denotes a new concept of multivariate $\varphi-$variation in the sense of Tonelli.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
