Interpolation and Approximation by Splines on [0, ∞ )

We investigate interpolation and approximation problems by splines, which possess a countable set of knots on the positive axis. In particular, we characterize those sets of points, which admit unique Lagrange interpolation and give some sufficient and some necessary conditions for best approximations. Moreover, we show that the classical results of spline-approximation theory are not available for splines with a countable set of knots.