Fourier-Bessel series for second-order and fourth-order Bessel differential equations
The authors consider the Hilbert function space framework for Fourier-Bessel series, based on linear differential operators generated by the second-order Bessel differential equation for Bessel functions of order zero \(-(xy'(x))'=\lambda x y(x)\), \(x\in(0,1]\), \(\lambda\in\mathbb C\) [see \textit{M. A. Naimark}, Linear differential operators. Part II, New York: Frederick Ungar Publishing Co. (1968; Zbl 0227.34020)] and by the fourth-order Bessel-type differential equation \[ (xy''(x))''-((9x^{-1}+8M^{-1}x)y'(x))'=\Lambda x y(x),\quad x \in(0,1],\tag{*} \] where \(M>0\), \(\lambda\in\mathbb C\) is the spectral parameter. In both cases the analysis is concerned with the theory of unbounded linear differential operators in the Hilbert function space \(L^2((0,1);x)\). It is proved that for \(f\in L^2((0,1);x)\) the Fourier-Bessel series \(f(x)=\sum_{m=1}^\infty a_mJ_0(j_mx)\), where \(j_m\) are positive zeros of \(J_0(x)\) and \(a_m=2J^{-2}_1(j_m) \int_0^1tf(t)J_0(j_mt)\,dt\) converges to \(f\) in \(L^2((0,1);x)\). A similar result is obtained for solutions to the fourth-order Bessel-type differential equation (*).
scientific article; zbMATH DE number 5305554
scientific article