Shape-Preserving C 1 and C 2 Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation

Francesc Aràndiga () and Sara Remogna
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Francesc Aràndiga: Departament de Matemàtiques, Universitat de València, Av. Vicent Andrés Estellés, E-46100 Burjassot, Spain
Sara Remogna: Department of Mathematics “G. Peano”, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy

Mathematics, 2025, vol. 13, issue 8, 1-20

Abstract: This paper addresses fundamental challenges in numerical approximation methods, focusing on balancing accuracy with shape-preserving properties. We present novel approaches that combine traditional spline methods with modern numerical techniques, extending existing quasi-interpolation techniques based on B-splines. Our methods maintain computational efficiency while better handling discontinuities, achieving C 1 and C 2 reconstructions and preserving essential shape properties. We demonstrate theoretical frameworks showing optimal approximation order O ( h d + 1 ) , d = 2 , 3 , with local reconstruction. Numerical experiments confirm significant improvements in accuracy and smoothness near discontinuities compared to existing methods, particularly in image processing and shock-capturing applications.

Keywords: spline quasi-interpolation; WENO; piecewise smooth functions approximation; shape-preserving (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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