Shape-Preserving C1 and C2 Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation - Francesc Aràndiga; Sara Remogna (Mathematics)
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 <semantics>C1</semantics> and <semantics>C2</semantics> reconstructions and preserving essential shape properties. We demonstrate theoretical frameworks showing optimal approximation order <semantics>O(hd+1)</semantics>, <semantics>d=2,3</semantics>, 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.