arXiv subjects

Giacomo Cappellazzo

Publications and source records attributed to Giacomo Cappellazzo.

3 recordsLinked to original sources

arXiv2026

Fast-Decaying Polynomial Reproduction

Polynomial reproduction plays a crucial role in deriving error estimates for various approximation schemes. In particular, local polynomial reproduction is a key ingredient in both error estimation and stability analysis. However, for certain computationally relevant methods, such as Rescaled Localized Radial Basis Functions (RL-RBF), this requirement constitutes a limitation. To enable the analysis of a broader class of approximation methods in a unified and efficient manner, the present work introduces a framework based on fast-decaying polynomial reproduction. In this approach, we do not restrict ourselves to compactly supported basis functions. Instead, we allow the basis functions to decay to zero at infinity, with the decay rate controlled as a function of the separation distance. The adoption of fast-decaying polynomial reproduction yields stable and convergent approximation schemes. These methods can achieve smoothness when used in conjunction with moving least squares. All theoretical results presented in this paper regarding the rate of convergence, the Lebesgue constant and the smoothness of the approximant have been numerically validated, including in the multivariate setting.

math.NA↗

arXiv2026

Scattered Data Histopolation in Averaging Kernel Hilbert Spaces

Kernel-based methods offer a powerful and flexible mathematical framework for addressing histopolation problems. In histopolation, the available input data does not consist of pointwise function samples but of averages taken over intervals or higher-dimensional regions, and these mean values serve as a basis for reconstructing or approximating the target function. While classical interpolation requires continuity of the underlying function, histopolation can be performed in larger function spaces. In the framework of kernel methods, we will introduce and study the so-called averaging kernel Hilbert spaces (AKHS's) for this purpose. Within this setting, we develop systematic construction principles for averaging kernels and provide characterizations based on the Fourier-Plancherel transform. In addition, we analyze several representative histopolation scenarios in order to highlight properties of this approximation method, including conditions for unisolvence and possible error estimates. Finally, we present numerical experiments that shed some light on the convergence behavior of the presented approach and demonstrate its practical effectiveness.

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arXiv2022

On Kosloff Tal-Ezer Least-Squares Quadrature Formulas

In this work, we study a global quadrature scheme for analytic functions on compact intervals based on function values on quasi-uniform grids of quadrature nodes. In practice it is not always possible to sample functions at optimal nodes that give well-conditioned and quickly converging interpolatory quadrature rules at the same time. Therefore, we go beyond classical interpolatory quadrature by lowering the degree of the polynomial approximant and by applying auxiliary mapping functions that map the original quadrature nodes to more suitable fake nodes. More precisely, we investigate the combination of the Kosloff Tal-Ezer map and least-squares approximation (KTL) for numerical quadrature: a careful selection of the mapping parameter ensures stability of the scheme, a high accuracy of the approximation and, at the same time, an asymptotically optimal ratio between the degree of the polynomial and the spacing of the grid. We will investigate the properties of this KTL quadrature and focus on the symmetry of the quadrature weights, the limit relations for the mapping parameter, as well as the computation of the quadrature weights in the standard monomial and in the Chebyshev bases with help of a cosine transform. Numerical tests on equispaced nodes show that a static choice of the map's parameter improve the results of the composite trapezoidal rule, while a dynamic approach achieves larger stability and faster convergence, even when the sampling nodes are perturbed. From a computational point of view the proposed method is practical and can be implemented in a simple and efficient way.

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