Rényi dimension and Gaussian filtering (Q997837)

This paper studies the upper and lower Rényi dimensions of a finite Borel measure \(\mu\) on \(\mathbb{R}^d\) \[ D_q^{+} (\mu) = {\lim_{\varepsilon\rightarrow 0}} \sup \frac{1}{q-1} \frac{\ln(S_{\mu}^q(\varepsilon))}{\ln(\varepsilon)} ~\text{and}~ D_q^{-} (\mu) = {\lim_{\varepsilon\rightarrow 0}} \inf \frac{1}{q-1} \frac{\ln(S_{\mu}^q(\varepsilon))}{\ln(\varepsilon)}, \] where \(S_{\mu}^q(\varepsilon) = \sum_{\mathbf{k}\in\mathbb{Z}^d} \mu (\varepsilon \mathbf{k} + \varepsilon \mathbb{I})^q\). Here \(\mathbb{I}\) is the product of \(d\) copies of \([0,1)\). The author revises \textit{C.-A. Guérin}'s result [J. Math. Phys. 42, No. 12, 5871--5875 (2001; Zbl 1008.28006)] by showing that if \(1< q < \infty\), or if \(0<q<1\) and \(\mu\) is \(q\)-finite, i.e., \(S_{\mu}^q(1) < \infty\), the Rényi dimension \(D_q^{\pm} (\mu)\) can be written in terms of the \(L^q\)-norms of convolutions of the Borel measure \(\mu\) against an approximate identity of Gaussians on \(\mathbb{R}^d\). Using this result, he obtains a Lipschitz-type estimation on the partition function \(S_{\mu}^q(\varepsilon)\) and apply it to generalize \textit{R. Riedi}'s result [J. Math. Anal. Appl. 189, No. 2, 462--490 (1995; Zbl 0819.28008)] by showing that the Rényi dimension can be computed as the discrete limits for sequences \(\{\varepsilon_n\}\) such that \(\varepsilon_n \searrow 0\) and \(\displaystyle\lim_{n\rightarrow\infty} \frac{\ln (\varepsilon_{n+1})}{\ln(\varepsilon_{n})} = 1\). He constructs a finite Borel measure \(\mu\) whose Rényi dimension cannot be computed by any sequence \(\{\varepsilon_n\}\) such that \(\varepsilon_n \searrow 0\) and \(\displaystyle\lim_{n\rightarrow\infty} \frac{\ln (\varepsilon_{n+1})}{\ln(\varepsilon_{n})} > 1\). Moreover, he obtains a bound relating the Rényi dimension \(D_q^{\pm} (\mu\ast\nu)\) of the convolution \(\mu\ast\nu\) with \(D_q^{\pm} (\mu)\) and \(D_q^{\pm} (\nu)\), which is different from the bounds obtained by \textit{J.-M. Barbaroux, F. Germinet}, and \textit{S. Tcheremchantsev} [J. Math. Pures Appl., IX. Sér. 80, No. 10, 977--1012 (2001; Zbl 1050.28006)]. Finally he gives an example which shows that the slopes of of the least-square best fit linear approximations to the partition function cannot always be used to calculate the Rényi dimension.