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A C∞ rational quasi-interpolation operator for functions with jumps without the Gibbs phenomenon

Author

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  • Dell’Accio, Francesco
  • Larosa, Francesco
  • Nudo, Federico
  • Siar, Najoua

Abstract

The study of quasi-interpolation has gained significant importance in numerical analysis and approximation theory due to its versatile applications in scientific and engineering fields. This technique provides a flexible and efficient alternative to traditional interpolation methods by approximating data points without requiring the approximated function to pass exactly through them. This approach is particularly valuable for handling jump discontinuities, where classical interpolation methods often fail due to the Gibbs phenomenon. These discontinuities are common in practical scenarios such as signal processing and computational physics. In this paper, we present a C∞ rational quasi-interpolation operator designed to effectively approximate functions with jump discontinuities while minimizing the issues typically associated with traditional interpolation methods.

Suggested Citation

  • Dell’Accio, Francesco & Larosa, Francesco & Nudo, Federico & Siar, Najoua, 2026. "A C∞ rational quasi-interpolation operator for functions with jumps without the Gibbs phenomenon," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 241(PA), pages 391-407.
    • Handle: RePEc:eee:matcom:v:241:y:2026:i:pa:p:391-407
    DOI: 10.1016/j.matcom.2025.09.004
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    References listed on IDEAS

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    1. Aràndiga, F. & Barrera, D. & Eddargani, S., 2024. "ENO and WENO cubic quasi-interpolating splines in Bernstein–Bézier form," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 225(C), pages 513-527.
    2. Aràndiga, F. & Barrera, D. & Eddargani, S. & Ibáñez, M.J. & Roldán, J.B., 2024. "Non-uniform WENO-based quasi-interpolating splines from the Bernstein–Bézier representation and applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 158-170.
    3. Foucher, Françoise & Sablonnière, Paul, 2009. "Quadratic spline quasi-interpolants and collocation methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(12), pages 3455-3465.
    4. Buhmann, Martin & Jäger, Janin & Jódar, Joaquín & Rodríguez, Miguel L., 2024. "New methods for quasi-interpolation approximations: Resolution of odd-degree singularities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 50-64.
    5. Barrera, D. & Eddargani, S. & Ibáñez, M.J. & Remogna, S., 2023. "Low-degree spline quasi-interpolants in the Bernstein basis," Applied Mathematics and Computation, Elsevier, vol. 457(C).
    6. Guessab, A. & Ibáñez, M.J. & Nouisser, O., 2011. "Error analysis for a non-standard class of differential quasi-interpolants," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(10), pages 2190-2200.
    7. Dell’Accio, Francesco & Di Tommaso, Filomena & Hormann, Kai, 2018. "Reconstruction of a function from Hermite–Birkhoff data," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 51-69.
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