A straightforward extension of some algorithms for solving unconstrained linear approximation problems in the L1 and L∞ norms is given. The extended algorithms allow lower or upper bounds to be placed on the parameters of the approximating function, whilst still retaining the computational efficiency of the unconstrained algorithm.nA representation of a cubic spline in terms of the values of its second derivatives at the knots and its values at the ends of the range is derived. By placing simple non-negativity or non-positivity constraints upon the values of these derivatives the spline can be forced to satisfy prescribed properties such as local convexity or concavity.nThe extended linear approximation algorithms, when used in conjunction with this representation of a cubic spline, enable approximations to discrete data sets to be obtained which are free from undesirable inflexions or oscillations.
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