Recursive construction of smoothing spline surfaces using normalized uniform B-splines

This paper considers the problem of constructing smoothing spline surfaces recursively each time when a new set of data is observed. The spline surfaces are constituted by employing normalized uniform B-splines as the basis functions. Then, based on the basic problem of optimal smoothing splines and an idea of recursive least squares method, we develop a recursive design algorithm of optimal smoothing splines. In addition, by assuming that the data for smoothing is obtained by sampling a surface, we analyze asymptotical and statistical properties of designed smoothing spline surfaces when the number of iterations tends to infinity. The algorithm and analyses are extended to the periodic case. We demonstrate the effectiveness and usefulness by numerical examples.