A selection of algorithms for the rational approximation matrix-valued functions are discussed, including variants interpolatory adaptive Antoulas--Anderson (AAA) method, Kry...

Abstract. We propose the application of hyperbolic interpolatory wavelets for large-scale -dimensional data fitting. In particular, we show how wavelets can be used as a highly efficient tool for multidimensional smoothing. The grid underlying these wavelets is a sparse grid. The hyperbolic interpolatory wavelet space of level uses basis functions and it is shown that under sufficient smoothnes...

In 1996, we constructed periodic interpolatory scaling functions φj , wavelet functions Lj and their dual basis φ̃j and L̃j with properties such as symmetry, biorthogonality, any order of smoothness, real-valuedness, explicit expressions and interpolatory. We proved the localization of φj in 1997, and in 1998 with Li proved the localization of Lj . In this paper we shall give a detailed proof of ...

For any interpolatory ternary subdivision scheme with two-ring stencils for a regular triangular or quadrilateral mesh, we show that the critical Hölder smoothness exponent of its basis function cannot exceed log3 11(≈ 2.18266), where the critical Hölder smoothness exponent of a function f : R2 → R is defined to be ν∞(f) := sup{ν : f ∈ Lip ν}. On the other hand, for both regular triangular and ...

We show the convergence (for all input data) of refinement rules in Riemannian manifolds which are analogous to the linear four-point scheme and similar univariate interpolatory schemes, and which are generalized to the Riemannian setting by the so-called log/exp analogy. For this purpose we use a lemma on the Hölder regularity of limits of contractive refinement schemes in metric spaces. In co...

In design of optical systems based on LED (Light emitting diode) technology, a crucial task is to handle the unstructured data describing properties of optical elements in standard formats. This leads to the problem of data fitting within an appropriate model. Newtons method is used as an upgrade of previously developed most promising discrete optimization heuristics showing improvement of both...

The quad/triangular subdivision, whose control net and refined meshes consist of both quads and triangles, provides better visual quality of subdivision surfaces. While some theoretical results such as polynomial reproduction and smoothness analysis of quad/triangle schemes have been obtained in the literature, some issues such as the basis functions at quad/triangle vertices and design of inte...

In this study, based on the optimal free derivative without memory methods proposed by Cordero et al. [A. Cordero, J.L. Hueso, E. Martinez, J.R. Torregrosa, Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation, Mathematical and Computer Modeling. 57 (2013) 1950-1956], we develop two new iterative with memory methods for solving a nonline...

Multiscale transforms for real-valued data, based on interpolatory subdivision operators, have been studied in recent years. They are easy to define and can be extended other types of example, manifold-valued data. In this chapter, we linear multiscale transforms, certain linear, non-interpolatory termed “even-reversible.” For such prove, using Wiener’s lemma, the existence an inverse operator ...