Orthogonal splines based on B-splines — with applications to least squares, smoothing and regularisation problems

Orthogonal linear and cubic splines are introduced, based on a simple recurrence procedure using B-splines. Stable formulae are obtained for explicit least squares approximation. An application to the smoothing of noisy data is given, in which the approximation is essentially a second integral of an orthogonal linear spline and this leads to an efficient solution procedure. An application to the regularisation of integral transforms, and specifically to the finite moment problem, is also described.