Spectral analysis and synthesis on symmetric spaces
The starting point of this paper is the Schwarz theorem which states that every non-trivial closed and translation-invariant subspace V of the space \(C^{\infty}(R)\) of smooth functions on the real line contains some function of the form \(p(x)e^{qx}\), where p(.) is a polynomial and q a complex number and moreover that the space V is spanned by such functions. The author studies the problem of Schwarz in the space \(C^{\infty}(X)\), where X is a rank one symmetric space of non-compact type and proves that the spectral analysis and synthesis hold for those invariant subspaces of \(C^{\infty}(X)\) whose spectrum does not contain the zeroes of the Harish-Chandra function c.
scientific article; zbMATH DE number 4056268
scientific article