Fast scattered data approximation with Neumann and other boundary conditions Original Research Article

An important problem in applications, such as signal and image procesing, is the approximation of a function f from a finite set of randomly scattered data f(xj). A common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials {e2πikx}. This leads to fast numerical algorithms, but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data which is rarely satisfied in practice. To overcome this drawback we impose Neumann boundary conditions on the data. This implies the use of cosine polynomials cos(πkx) as basis functions. We show that using cosine polynomials leads to a least squares problem involving certain Toeplitz-plus-Hankel matrices and derive estimates on the condition number of these matrices. Unlike other Toeplitz-plus-Hankel matrices, these matrices cannot be diagonalized by the discrete cosine transform (DCT), but they still allow a fast matrix–vector multiplication via DCT which gives rise to fast conjugate gradient type algorithms. We show how the results can be generalized to higher dimensions. We also consider anti-symmetric boundary conditions, leading to sine polynomials as proper trigonometric basis. Finally we demonstrate the performance of the proposed methods by an application to a two-dimensional geophysical scattered data problem.