  We discuss two topics related to Fourier transforms on Lie groups and on
homogeneous spaces: the operational calculus and the Gelfand--Gindikin problem
(program) about separation of non-uniform spectra. Our purpose is to indicate
some non-solved problems of non-commutative harmonic analysis that definitely
are solvable. This is a sketch of my talks on VI School "Geometry and Physics",
Bialowieza, Poland, June 2017.
