Fourier series with nonnegative coefficients on compact semisimple Lie groups
Consider a compact semi-simple Lie group U with Lie algebra \({\mathfrak u}=Lie U\) and a Cartan decomposition \({\mathfrak u}={\mathfrak k}\oplus {\mathfrak p}\) with respect to a fixed Cartan involution \(\theta\). Let K be the analytic subgroup of U with Lie algebra \({\mathfrak k}\). Normalizing the Haar measure dk by the condition \(\int_{K}dk=1\), one defines the spherical function \(\psi_{\lambda}\) on U corresponding to \(\lambda \in U_ K^{{\hat{\;}}}\), the set of all equivalence classes of irreducible representations of U of class one with respect to K, via the characters \(\chi_{\lambda}\), \(\psi_{\lambda}(u):=\int_{K}\chi_{\lambda}(u^{-1}k)dk\). Then the Fourier expansion of f in the space \(L^ 1(U//K)\) of K-biinvariant \(L^ 1\) functions on U is given as \[ f(u)=\sum_{\lambda \in U_ K^{{\hat{\;}}}}d_{\lambda}f^{{\hat{\;}}}(\lambda)\psi_{\lambda}(u),\text{ where } \hat f(\lambda)=\int_{U}f(u)\chi_{\lambda}(u^{-1})du. \] The author proves the \(L^ q\) behavior, \(q=p/(p-1)\), \(1<p\leq 2\), of the Fourier coefficients, \[ \| f^ 1\|_{\#,q}=(\sum_{\lambda \in U^{{\hat{\;}}}}d_{\lambda}^{2-q}| f^{{\hat{\;}}}(\lambda)|^ q)^{1/q}<\infty \] for the central functions \(f\in L^ 1_{\#}(U)\) having nonnegative Fourier coefficients and the cut-off function \(f_{\Xi_{\#}}\) which coincides with f on \(\Xi_{\#}\) and vanishes outside \(\Xi_{\#}\), is of class \(L^ p(U)\), where \(\Xi_{\#}:=\cup_{u\in G}u\Xi u^{-1}\) for some neighborhood \(\Xi\) of unity in U. If the zonal function \(f\in L^ 1(U//K)\) has nonnegative Fourier coefficients and if \(f_{\Xi_ b}\in L^ p(U)\), where \(\Xi_ b:=K\Xi K\), then \[ \| \hat f\|_{b,p}:=(\sum_{\lambda \in U_ K^{{\hat{\;}}}}d_{\lambda}| \hat f(\lambda)|^ q)^{1/q}<\infty. \] Their proof uses the technique used by \textit{J. Ash}, \textit{M. Rains} and \textit{S. Vági} [Proc. Am. Math. Soc. 101, 392- 393 (1987; Zbl 0653.42010)].
scientific article; zbMATH DE number 4205106