Bessel wavelets and the Galerkin analysis of the Bessel operator (Q5949589)

The authors consider the Bessel operator, which is the self-adjoint Friedrichs extension of the operator \(\widetilde L (g) (x) = - g''(x)+ \frac{\nu^2-1/4}{x^2}g(x)\), for \(f \in C_0^\infty(\mathbb R^+)\), \(x\in {\mathbb R}^+\) and \(\nu \geq 0\). Their goal is to study Galerkin approximations to the solution of the equation \(Lu=f\) for some given \(f\). The approximation is in the form of a representation in a Bessel wavelet, i.e., a collection of functions generated by a single function \(\psi \in L^2(\mathbb R^+)\) as follows: NEWLINE\[NEWLINE\psi_{j,k} (x) = (2\pi)^{-1/2}2^{-j/2+1}\sqrt{x} \int_0^\infty \sin(2^{-j} (k-1/2)\xi) w(2^{-j}\xi) J_\nu(x\xi) \sqrt{\xi} d\xi,NEWLINE\]NEWLINE for \((j,k) \in {\mathbb Z}\times {\mathbb N}\), where \(J_\nu\) is the Bessel function of order \(\nu\), and \(w\) - an even, smooth, compactly supported, real-valued function on \(\mathbb R\). These functions form an orthonormal basis for \(L^2(\mathbb R^+)\). One may consider projections of the solutions onto the subspaces generated by functions \(\{\psi_{j,k}: (j,k) \in \Lambda \subset {\mathbb Z}\times {\mathbb N}\}\) for any \(\Lambda\). NEWLINENEWLINENEWLINEThe authors show that there exist diagonal preconditioners for the discrete systems associated with the above subspaces, which have condition numbers bounded independent of the subspace. Moreover, such systems are sparse and the condition numbers are small enough for practical applications.