Lobachevsky spline functions and interpolation to scattered data

Multidimensional Lobachevsky Spline Integration on Scattered Data

“…Finally, we observe that, since Lobachevsky splines are (univariate) strictly positive definite functions for even n ≥ 2, we can construct multivariate strictly positive definite functions from univariate ones (see, e.g., [25]), expressing them as products of Lobachevsky splines [4].…”

“…Note that a numerically equivalent technique, despite computationally less efficient, to evaluate Lobachevsky spline integrals is given in [5].…”

“…In particular, we consider the problem of constructing new integration formulas for high-dimensional scattered data by a class of spline functions, called Lobachevsky splines, arisen in probability theory [17,21] and then also proposed in multivariate interpolation on scattered data [4] and landmark-based image registration [1,3,6]. Lobachevsky splines consist in an infinite sequence of univariate spline functions depending on a shape parameter, which are compactly supported, strictly positive definite, and enjoy noteworthy theoretical and computational properties, such as the convergence to the Gaussian function and the convergence of the sequence of their integrals and derivatives to integrals and derivatives of the Gaussian, respectively (see [4]).…”

“…Lobachevsky splines consist in an infinite sequence of univariate spline functions depending on a shape parameter, which are compactly supported, strictly positive definite, and enjoy noteworthy theoretical and computational properties, such as the convergence to the Gaussian function and the convergence of the sequence of their integrals and derivatives to integrals and derivatives of the Gaussian, respectively (see [4]). Now, starting from the previous work [5], where we investigate the integration problem on scattered data in R d , for d = 1, 2, in this paper we focus on the use of Lobachevsky spline integration formulas for 3 ≤ d ≤ 10. The idea of using Lobachevsky splines to construct integration formulas on scattered data turns out to be quite natural, not only for the recent interest in meshfree numerical integration, but also for the performance and computation easiness of Lobachevsky spline integrals.…”

“…Moreover, we remark that Lobachevsky spline interpolation formulas are neither mesh-based formulas (no grid is here considered) nor radial ones, but they asymptotically behave like Gaussian interpolants (see [4]). This feature, together with the uniform convergence of Lobachevsky splines to the Gaussians, allows us to give error estimates for Lobachevsky splines integration formulas.…”