Spectral synthesis on the group of conformal automorphisms of the unit disc. (English. Russian original) Zbl 1400.43002
Sb. Math. 209, No. 1, 1-34 (2018); translation from Mat. Sb. 209, No. 1, 3-36 (2018).
Let \(G\) be the Lie group of conformal automorphisms of the unit disc \({\mathbb D}=\{z\in {\mathbb C} : |z|<1\}\), and let \(K=\mathrm{SO}(2)\) be the subgroup of \(G\) that consists of all rotations of the plane \({\mathbb C}\). Let \({\mathcal U}\) be an arbitrary nontrivial closed linear subspace of the space \({\mathcal E}=C^\infty(G)\) which is invariant under shifts \(f(g)\mapsto f(kgh)\), \(k\in K\), \(h\in G\). The central result of the paper is the theorem that such subspace \({\mathcal U}\) admits spectral synthesis, that is \({\mathcal U}\) coincides with a closed linear span of certain special functions on \(G\).
Reviewer: Sergei S. Platonov (Petrozavodsk)
MSC:
| 43A45 | Spectral synthesis on groups, semigroups, etc. |
| 43A85 | Harmonic analysis on homogeneous spaces |
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