Scattered data approximation by positive definite kernel functions (Q2898489)

The author presents selected aspects of kernel-based scattered data approximation.NEWLINENEWLINEConcerning Lagrange interpolation, since for any distinct points \(X=\{x_1,\ldots,x_n\}\) the Vandermonde matrix is non singular only in the univariate case, in general for the multivariate case this is not assured. This means that the basis of the \(n\)-dimensional linear function space, where the interpolant of an unknown function \(f\) is looked for, must necessarily depend on the interpolation points \(X\), i.e., it is of the kind \(\{s_j=K(\cdot,x_j),\;1\leq j\leq n\}\). The condition for the Vandermonde matrix to be non singular for all choices of \(X\) is satisfied if \(K(x,y)\) is a symmetric and positive definite function. Then the author describes how to contruct positive definite functions by using the Fourier transform and by convolutions, listing properties and some examples. This ideas have led to the construction of compactly supported positive definite (radial) functions.NEWLINENEWLINESuch a positive definite function \(K\) is also the unique reproducing kernel of its associated Hilbert space, also known as native space of \(K\). Properties of the native space and some examples are discussed.NEWLINENEWLINEThen some approximation properties of this scattered data reconstruction method are presented, showing the optimality of the positive definite kernel-based interpolation scheme with respect to energy minimization, to best approximation and to norm minimization of the pointwise error functionals.NEWLINENEWLINEMoreover, the conditioning of the interpolation problem and the stability of the recovery method are analyzed, providing also some bounds for the associated Lebesgue constant.NEWLINENEWLINEThe last part of the paper is devoted to penalized least squares approximation, an alternative approach for scattered data fitting other than Lagrange interpolation. This kind of approximation is useful for either very large data sets or for data contamined with noise. The well-posedness and the sensitivity of the problem, the unicity and the characterization of the solution and its convergence to the solution of the classical least squares approximation are discussed.