The restrictions of functions holomorphic in a domain to curves lying on its boundary, and discrete $\operatorname{SL}_2(\mathbb R)$-spectra

We consider the operator of restriction of functions holomorphic in a ball or a polydisc to curves lying on the Shilov boundary. It turns out that any function with polynomial growth near the boundary has such a restriction if the position of the curve satisfies a certain condition: if the domain is a ball, then the curve must be transversal to the standard contact distribution on the sphere, and if the domain is a polydisc, then the curve must be monotonic increasing with respect to all coordinates in the standard coordinatization of the torus. We use assertions of this kind to obtain a simple description of discrete inclusions in spectra (of minimal invariant subspaces) for several problems of $\operatorname{SL}_2(\mathbb R)$-harmonic analysis.