Self-similarity in harmonic analysis
The author surveys his recent work on harmonic analysis and self-similarity. He replaces Lebesgue measure by self-similar measures. The main tool is the Fourier transform of such a measure. For example, the Cantor measure \(\mu = {1\over 2} \mu \circ S^{-1}_0 + {1\over 2} \mu \circ S^{-1}_1\), where \(S_0(x) = {1\over 3} x\) and \(S_1(x) = {1\over 3} x + {2\over 3} ,\) has a Fourier transform \(F\) with the infinite product representation \[ F(\xi) = e^{i\xi/3} \prod^\infty_{k = 1} \cos \xi/3^k. \] He discusses the fractal Plancherel theorem, convolutions, certain dimensions, wavelets, the fractal Radon inversion problem and connections to self-similar tilings of the \(\mathbb{R}^n\). Finally, the results are embedded in the context of stratified nilpotent Lie groups. Many open problems are explicitly mentioned.
scientific article; zbMATH DE number 846177