Accelerating Bernstein approximation by Lagrange-based hybridization

Duman O., DUMAN E.

Mathematics and Computers in Simulation, cilt.250, ss.524-539, 2026 (SCI-Expanded, Scopus)

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 250
  • Basım Tarihi: 2026
  • Doi Numarası: 10.1016/j.matcom.2026.07.004
  • Dergi Adı: Mathematics and Computers in Simulation
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Applied Science & Technology Source, Compendex, INSPEC, MathSciNet, Public Affairs Index, zbMATH, Academic Search Ultimate (EBSCO), Engineering Source (EBSCO)
  • Sayfa Sayıları: ss.524-539
  • Anahtar Kelimeler: Bernstein polynomials, Lagrange interpolation, Max-product bernstein operators
  • Gazi Üniversitesi Adresli: Evet

Özet

In this paper, we develop a Lagrange-based hybridization technique for accelerating Bernstein-type approximation. The main idea is to replace the point samples in Bernstein-type schemes by fixed-degree local Lagrange interpolants constructed on clamped sliding windows of equispaced nodes, and then to aggregate these local polynomials by Bernstein weights. In this way, the proposed operators retain the stabilizing effect and endpoint control of Bernstein-type averaging while incorporating higher-order local information. We study both a linear Bernstein-Lagrange hybrid operator and its nonlinear max-product counterpart. For sufficiently smooth functions, we establish improved uniform approximation rates, showing that the convergence order increases with the degree of the local interpolation. The construction is further extended to several variables through tensor-product local interpolation, where the approximation behavior is governed by the smallest local degree among the coordinate directions. Numerical experiments illustrate the improvement over the classical Bernstein and max-product Bernstein operators and demonstrate the suppression of Runge-type oscillations. Finally, a level-set-based geometric application is presented, showing how the multivariate hybrid operators can be used for implicit surface approximation and Hausdorff convergence of symmetric thickened zero level sets.