This dissertation gives explicit algorithms for constructing multiple types of high dimen-sional 1D splines (standard and exponential A-splines, standard and exponential C-splines), and generating L2-orthogonal bases for various families of splines (via the standard and ex- ponential A-spline procedures). These orthogonal bases of spline functions are used in L2 approximation of functions by way of orthogonal projection, and relevant error bounds for these approximations are given in L 2 and L ∞ . The 1D spline approximation procedures developed here are used in construction of tensor product approximations of multivariate functions. Computational examples are provided.