Completeness and minimality of systems of Bessel functions

Abstract

|= +∞.At studying of some non-classical boundary-value problems (see [26]–[31]) and generalizedeigenvectors of some linear operators [28], [29] we needed to obtain the analogues of Theorems 1-3 for weighted spaces and establish an approximation properties of the special finite linearcombinations of Bessel functions. We don’t understand to the end the nature of expectedresults for an arbitrary ν ∈ R. For advance in the given direction it is important to investigatein details the simplest model cases ν = −3/2 and ν = 3/2. The case ν = −3/2 was consideredin [27], [30] (see also [26], [28], [31]). Here we consider the case ν = 3/2 more detail. But even inthis case we cannot obtain the all necessary facts. In particular, remains an open one for us theproblem formulated at the end of this paper. In our view, its solution is very important for theconstruction of some spectral theory that is based on the notion of a generalized eigenvector(see [28], [29]).It is well known (see [3], [25, p. 350], [32]) that√zJ