Order-preserving derivative approximation with periodic radial basis functions

8 years 13 days ago

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In this exploratory paper we study the convergence rates of an iterated method for approximating derivatives of periodic functions using radial basis function (RBF) interpolation. Given a target function sampled on some node set, an approximation of the mth derivative is obtained by m successive applications of the operator “interpolate, then diﬀerentiate” - this process is known in the spline community as successive splines or iterated splines. For uniformly spaced nodes on the circle, we give a suﬃcient condition on the RBF kernel to guarantee that, when the error is measured only at the nodes, this iterated method approximates all derivatives with the same rate of convergence. We show that thin-plate spline, power function, and Mat´ern kernels restricted to the circle all satisfy this condition, and numerical evidence is provided to show that this phenomena occurs for some other popular RBF kernels. Finally, we consider possible extensions to higher-dimensional periodic dom...

Edward J. Fuselier, Grady B. Wright

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