  We consider the problem of estimating an unknown function f* and its partial
derivatives from a noisy data set of n observations, where we make no
assumptions about f* except that it is smooth in the sense that it has square
integrable partial derivatives of order m. A natural candidate for the
estimator of f* in such a case is the best fit to the data set that satisfies a
certain smoothness condition. This estimator can be seen as a least squares
estimator subject to an upper bound on some measure of smoothness. Another
useful estimator is the one that minimizes the degree of smoothness subject to
an upper bound on the average of squared errors. We prove that these two
estimators are computable as solutions to quadratic programs, establish the
consistency of these estimators and their partial derivatives, and study the
convergence rate as n increases to infinity. The effectiveness of the
estimators is illustrated numerically in a setting where the value of a stock
option and its second derivative are estimated as functions of the underlying
stock price.
