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Radial basis functions play an increasingly prominent role in modern approximation. They are widely used in scattered data fitting, numerical solution of partial differential equations, machine learning and others. Although radial basis functions have excellent approximation properties, they often produce highly ill-conditioned discrete algebraic system and lead to a high computational cost. The paper introduces local multilevel scattered data interpolation method, which employ nested scattered data sets and scaled compactly supported radial basis functions with varying support radii. We will provide convergence theory for Sobolev target functions. And several numerical experiments will be provided to conform the efficiency of new method. [ABSTRACT FROM AUTHOR]

The notion of cubic fractal interpolation function (FIF) has received considerable attention in the literature due to its versatility, flexibility and ease of implementation. In this article, we shall view cubic FIFs as a family of C 1 -continuous fractal functions associated with the traditional C 1 -continuous cubic spline. General theorems that identify suitable values of the parameters so as to constrain a fractal function and its first derivative within suitable axis-aligned rectangles are reported. By applying these theorems, cubic fractal interpolation of a data set subject to strip conditions on the interpolant and its first derivative is discussed. These results are applied to investigate positivity and monotonicity properties of a hybrid bivariate interpolant over a rectangular region R obtained by blending univariate cubic FIFs via bicubically blended Coons patch. The L ∞ -norm of the error in approximating a function f ∈ C 2 ( R ) with the proposed bivariate interpolant is shown to be of order O ( h 2 ) as h → 0. [ABSTRACT FROM AUTHOR]

In this paper we focus, from a mathematical point of view, on properties and performances of some local interpolation schemes for landmark-based image registration. Precisely, we consider modified Shepard's interpolants, Wendland's functions, and Lobachevsky splines. They are quite unlike each other, but all of them are compactly supported and enjoy interesting theoretical and computational properties. In particular, we point out some unusual forms of the considered functions. Finally, detailed numerical comparisons are given, considering also Gaussians and thin plate splines, which are really globally supported but widely used in applications. [ABSTRACT FROM AUTHOR]