Jacobi polynomial expansions
The paper is devoted to eigenfunction expansions associated with Jacobi polynomials \([^{\alpha,\beta} P_ n]^ \infty_{n=0}\). The classical expansion associated with the Jacobi operator is well-known when \(a>-1\), \(\beta> -1\). It is shown that when \(\alpha\), \(\beta<-1\), there are eigenfunction expansions associated with Jacobi operators and Jacobi polynomials, of course in a different function space which is a Sobolev space. It is shown that in this space, the Jacobi operator is self- adjoint. The eigenfunction expansions of the space remain valid for all \(\alpha\), \(\beta\) and are same as there in the \(L_ 2\)-space when \(\alpha\), \(\beta>-1\). Finally, it is shown that spectral resolution in the \(L_ 2\) space hold in the new space mentioned above.
scientific article; zbMATH DE number 567023