A nested Bernstein form approximation method by interval partitioning

doi:10.1016/j.jmaa.2025.130192

2025-10-01

PRE

Beong In Yun *

DOI：10.1016/j.jmaa.2025.130192

原文链接

原文求助

摘要

En 中文

For a real valued continuous function, we propose a Bernstein form approximation method based on dividing a given interval. We employ traditional Bernstein polynomials on the divided subintervals, and then use a sigmoid function to smoothly combine Bernstein polynomials at the matching endpoints of the subintervals. In general, for any integer p >= 1, we develop p-fold nested Bernstein form approximation method by repeating this approximation process. We prove the asymptotic uniform convergence of the proposed method and its derivatives for sufficiently large order of the sigmoid function used, as well as the asymptotic interpolation property at division nodes. The usefulness of the proposed method is demonstrated through numerical implementations on some selected test functions. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Keyword:

Bernstein polynomial

Weierstrass approximation theorem

Sigmoid function

Nested Bernstein form approximation

Interval partitioning