Surface fitting using convex tensor-product splines

A construction of linear sufficient convexity conditions for polynomial tensor-product spline functions is presented. As the main new feature of this construction, the obtained conditions are asymptotically necessary: increasing the number of linear inequalities in a suitable manner adapts them to any finite set of strongly convex spline surfaces. Based on the linear constraints we formulate least-squares approximation of scattered data by spline surfaces as a quadratic programming problem.