Hermite expansions on \({\mathbb{R}}^{2n}\) for radial functions
Let \(\Phi_{\nu}(x)\) denote the normalized Hermite functions on \({\mathbb{R}}^ n\) which are eigenfunctions of the Hermite operator \[ H_ n=(-\Delta +(1/4)| x|^ 2). \] Let \(P_ Nf\) denote the projection of \(L^ 2({\mathbb{R}}^ n)\) onto the space spanned by \(\{\Phi_{\nu}:\;| \nu | =N\}.\) Then the Riesz means of order \(\delta >0\) are defined by \[ S^{\delta}_ Lf=\sum (1- \frac{2N+n}{2L})^{\delta}_+ P_ Nf. \] The author establishes the following theorem. ``Let \(f\in L^ p({\mathbb{R}}^{2n})\) be radial, \(1<p<4n/(2n+1)\) and \(\delta >\delta (p).\) Then there is a C independent of L such that \(\| S^{\delta}_ Lf\| <C\| f\|_ p\) holds.''
scientific article; zbMATH DE number 4210794