Asymptotic expansions for Fourier transform of singular self-affine measures (Q1340551)

Let \(S_ 1,\dots, S_ N\) be a finite set of affine contractions on \(\mathbb{R}\) such that \(S_ k x=\lambda_ k x+ a_ k\) for \(k=1, \dots,N\). The author considers singular self-affine measures \(\mu\) which live on the invariant set \(M\) which is generated by the IFS \(\{\mathbb{R}, S_ 1,\dots, S_ N\}\). Let \(\kappa\) be the Hausdorff dimension of \(M\), \(\widehat{\mu}\) the Fourier transform of \(\mu\) and \[ m(a)= a^{\kappa-1} \int_{-a}^ a |\widehat {\mu}|^ 2 (p)dp. \] The function \(m(\cdot)\) has an asymptotic expansion by finite series involving some periodic functions as well as a function which has some tempered behavior at infinity. The paper is also concerned with the inverse reconstruction problem for the self-affine measures.