Fitting spline functions by the method of least squares

A spline function of degree k with knots S₀, S₁,...,Sr is a C[superscript]k-1 function which is a polynomial of degree at most k in each of the intervals (-∞, S₀), (S₀, S₁),…, (Sr,+∞). The Gauss-Markoff Theorem can be used to estimate by least squares the coefficients of a spline function of given degree and knots. Estimating a spline function of known knots without full knowledge of the degree entails an extension of the Gauss-Markoff technique. The estimation of the degree when the knots are also unknown has a possible solution in a method employing finite differences. The technique of minimizing sums of squared residuals forms the basis for a method of estimating the knots of a spline function of given degree. Estimates for the knots may also be obtained by a method of successive approximation, provided additional information about the spline function is known.Science, Faculty ofMathematics, Department ofGraduat