Large-scale renormalisation of Fourier transforms of self-similar measures and self-similarity of Riesz measures
Let \(f\) be a weak limit of \(f_n(\omega)= \prod^{n- 1}_{j= 0} (1+ \cos(3^j \omega))\). It is known as a \(2\pi\)-periodic positive measure. The author shows that in the sense of distributions \(\lim_{n\to \infty} 2^{- n} f(3^{- n}\omega)= f^*(\omega)\) and identifies \(f^*\) as a positive measure that is absolutely continuous with respect to \(f\), \[ {df^*(\omega)\over df(\omega)}= |\widehat \mu(\omega/2)|^2, \] where \(\widehat \mu\) is the Fourier transform of the Cantor measure.
scientific article; zbMATH DE number 897955