Stability Results for Scattered Data Interpolation by Trigonometric Polynomials

A fast and reliable algorithm for the optimal interpolation of scattered data on the torus $\mathbb{T}^d$ by multivariate trigonometric polynomials is presented. The algorithm is based on a variant of the conjugate gradient method in combination with the fast Fourier transforms for nonequispaced nodes. The main result is that under mild assumptions the total complexity for solving the interpolation problem at $M$ arbitrary nodes is of order ${\cal O}(M\log M)$. This result is obtained by the use of localized trigonometric kernels where the localization is chosen in accordance with the spatial dimension $d$. Numerical examples show the efficiency of the new algorithm.

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