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Discrete variable representation (DVR) basis sets on the radial half-line, based on Bessel functions, are presented. These are Hankel transforms of the eigenfunctions of the particle in a spherical box in k space, but there is no box or bound on the radial variable r. The grid points extend to infinity on the r axis. The DVR functions are exactly orthonormal and exactly satisfy the interpolation properties usually associated with DVR functions. The exact matrix elements of the kinetic energy are computed, and the use of the Bessel DVR functions in radial eigenvalue problems is illustrated. The phase space or semiclassical interpretation of the Bessel DVR functions is presented, and variations on these functions, corresponding to alternative boundary conditions in k space, are discussed. An interesting feature of Bessel DVR functions is that they are based on a finite basis representation that is continuously infinite. [ABSTRACT FROM AUTHOR]