Wavelet expansions of fractal measures
The author continues his studies in fractal spectral asymptotics of measures [see Bull. Am. Math. Soc., New Ser. 20, No.1, 55-59 (1989; Zbl 0668.28003), J. Funct. Anal. 89, No.1, 154-187 (1990; Zbl 0693.28005), Indiana Univ. Math. J. 39, No.3, 797-817 (1990; Zbl 0706.28008)] by means of wavelet expansions. A measure \(\mu\) on \({\mathbb{R}}^ n\) is said to be locally \(\alpha\)-dimensional if it satisfies \(\mu (B_ r(x))\leq cr^{\alpha}\) for all \(x\in {\mathbb{R}}^ n\) and all \(r\leq 1,\) where \(B_ r(x)\) denotes the ball of radius r at x. Such measures can be decomposed as \(\mu =\phi d\mu_{\alpha}+\nu,\) where \(\mu_{\alpha}\) is the \(\alpha\)-dimensional Hausdorff measure and \(\mu_{\alpha}(E)<\infty\) implies \(\nu (E)=0.\) If \(\psi^{(\epsilon)}_{j,k}(x)=2^{n_ j/2}\psi^{(\epsilon)}(2^ jx-k)\), \(x\in {\mathbb{R}}^ n\), \(j\in {\mathbb{Z}}\), \(k\in {\mathbb{Z}}^ n\) and \(\epsilon\in E\), E a finite set, denotes a wavelet basis with generators \(\psi^{(\epsilon)}\), the main result is then concerned with the projection operator \(P_ j\) at level j defined as \[ P_ j(f d\mu)(x)=\sum_{k\in {\mathbb{Z}}^ n,\epsilon \in E}<f d\mu,\psi^{(\epsilon)}_{j,k}>\psi^{(\epsilon)}_{j,k}(x),\quad x\in {\mathbb{R}}^ n, \] where \(<f d\mu,\psi^{(\epsilon)}_{j,k}>=\int f(x)\psi^{(\epsilon)}_{j,k}(x)d\mu (x)\) for a locally \(\alpha\)- dimensional measure \(\mu\) and \(f\in L^ p(\mu),\quad 1\leq p\leq \infty.\) It is proved for example that in \(L^ p\)-norm \[ \| P_ j(f d\mu)\| \leq c2^{j((n-\alpha)/p')}\| f\|, \] \[ \limsup_{j\to \infty}2^{j((n-\alpha)/p')}\| P_ j(f d\mu)\| \leq (\int | f|^ p\phi d\mu_{\alpha})^{1/p}. \]
fractal spectral asymptotics of measures
scientific article; zbMATH DE number 4214294
scientific article