Normalized system for wave and Dunkl operators
Normalized systems of functions [\textit{V. V. Karachik}, J. Math. Anal. Appl. 287, No. 2, 577--592 (2003; Zbl 1039.31009)] with respect to the wave and to the Dunkl operators are constructed. Let \([x,x]=x_1^2+\cdots+x_p^2-x_{p+1}^2-\cdots-x_{n}^2\) and \[ J_kf(x)=\int_0^1(1-t)^{k-1} t^{n/2-1}f(tx)\,dt. \] The wave operator is defined as \(\square u=u''_{x_1}+\cdots+ u''_{x_p}- u''_{x_{p+1}}-\cdots- u''_{x_n}\). In Theorem 3.1 it is proved that the system of functions \(G_0(x)=u(x)\), \(G_k(x)=1/(4^kk!(k-1)!) [x,x]^{2k}J_ku(x)\), where \(\square u(x)=0\) is 0-normalized with respect to the wave operator \(\square\), i.e., it is such that \(\square G_k(x)=G_{k-1}(x)\) and \(\square G_0(x)=0\). Using this system the Riquier's problem for the wave operator \(\square\) is solved in Theorem 3.3. With the help of the operator \(J_k\) a 0-normalized system of functions with respect to the Dunkl Laplacian \(\Delta_h=\sum_{j=1}^n\mathcal D^2_j\) is constructed in Theorem 4.1.
scientific article; zbMATH DE number 5788309