Box dimension and fractional integral of linear fractal interpolation functions
Let \(g\) be a continuous function on \([a,b]\subset\mathbb{R},\) and \(\Gamma _{g}=\{(x,g(x))\left| x\in[ a,b]\right. \}\) be the graph of \(g.\) For \(0<\varepsilon<(b-a)/2,\) let \(\{\tau_{\ell}\}_{\ell=0}^{m}\) be an \(\varepsilon\)-partition of \([a,b]\) (i.e. \(\tau_{0}=a,\) \(\tau_{m}=b,\) \(\varepsilon/2<\tau_{\ell+1}-\tau_{\ell}\leq\varepsilon,\) \(\ell =0,1,2,\dots,m-1)\) and let \(O(g,p,\varepsilon)\) be the oscillation of \(g\) on \([p,p+\varepsilon]\cap[ a,b],\) \(p\in\mathbb{R}\). Define \(\mathcal{N} _{\Gamma_{g}}^{\ast}(\varepsilon,\{\tau_{\ell}\})=\varepsilon^{-1} \sum_{\ell=0}^{m-1}O(g,\tau_{\ell},\varepsilon)\) and denote \(\mathcal{N}_{\Gamma_{g}}^{\ast}(\varepsilon)=\inf\{\mathcal{N}_{\Gamma_{g} }^{\ast}(\varepsilon,\{\tau_{\ell}\}\left| \{\tau_{\ell}\}\right. \) is an \(\varepsilon\)-partition of \([a,b]\}.\) In these terms, the authors define the box dimension of the graph \(\Gamma_{g}\) of a continuous function \(g\) and this box dimension is \(\dim_{B}(\Gamma_{g})=\max\left\{ 1,\lim_{\varepsilon \rightarrow0}\frac{\log\mathcal{N}_{\Gamma_{g}}^{\ast}(\varepsilon)}{\log \frac{1}{\varepsilon}}\right\} \) if the limit exists. If \(f\) is a linear FIF (fractal interpolation function; for definition see [\textit{M. F. Barnsley}, Constructive Approximation 2, 303--329 (1986Zbl 0606.41005)]) determined by \(\{L_{i}(x),\) \(F_{i}(x,y)\}_{i=1}^{N},\) where \(L_{i}(x)=a_{i}x+b_{i}\) and \(F_{i}(x,y)=d_{i}y+q_{i}(x)\) and one supposes that \(\sum_{i=1}^{N}\left| d_{i}\right| >1\) and \(\dim_{B} (\Gamma_{q_{i}})=1,\) \(1\leq i\leq N,\) then \(\dim_{B}(\Gamma_{g})=D(\{a_{i} ,d_{i}\})\) or \(1,\) where \(D(\{a_{i},d_{i}\})\) is the unique solution \(s\) of the equation \(\sum_{i=1}^{N}a_{i}^{s-1}\left| d_{i}\right| =1.\) The authors also prove that the fractional integral \(I_{x_{0}^{+}}^{v}f\) of a linear FIF \(f\) is also a linear FIF and give the box dimension of it, under some additional conditions.
scientific article; zbMATH DE number 5639039